5 research outputs found
Paired many-to-many 2-disjoint path cover of balanced hypercubes with faulty edges
As a variant of the well-known hypercube, the balanced hypercube was
proposed as a novel interconnection network topology for parallel computing. It
is known that is bipartite. Assume that and
are any two sets of two vertices in different partite sets of
(). It has been proved that there exist two vertex-disjoint
-path and -path of covering all vertices of it. In
this paper, we prove that there always exist two vertex-disjoint -path
and -path covering all vertices of with at most faulty
edges. The upper bound of edge faults tolerated is optimal.Comment: 30 pages, 9 figure
The restricted -connectivity of balanced hypercubes
The restricted -connectivity of a graph , denoted by , is
defined as the minimum cardinality of a set of vertices in , if exists,
whose removal disconnects and the minimum degree of each component of
is at least . In this paper, we study the restricted -connectivity of the
balanced hypercube and determine that
for . We also obtain a sharp upper
bound of and of -dimension balanced
hypercube for (). In particular, we show that
Structure and substructure connectivity of balanced hypercubes
The connectivity of a network directly signifies its reliability and
fault-tolerance. Structure and substructure connectivity are two novel
generalizations of the connectivity. Let be a subgraph of a connected graph
. The structure connectivity (resp. substructure connectivity) of ,
denoted by (resp. ), is defined to be the minimum
cardinality of a set of connected subgraphs in , if exists, whose
removal disconnects and each element of is isomorphic to (resp. a
subgraph of ). In this paper, we shall establish both and
of the balanced hypercube for
.Comment: arXiv admin note: text overlap with arXiv:1805.0846
Hamiltonian cycles in hypercubes with faulty edges
Szepietowski [A. Szepietowski, Hamiltonian cycles in hypercubes with
faulty edges, Information Sciences, 215 (2012) 75--82] observed that the
hypercube is not Hamiltonian if it contains a trap disconnected halfway.
A proper subgraph is disconnected halfway if at least half of its nodes
have parity 0 (or 1, resp.) and the edges joining all nodes of parity 0 (or 1,
resp.) in with nodes outside , are faulty. The simplest examples of such
traps are: (1) a vertex with incident faulty edges, or (2) a cycle
, where all edges going out of the cycle from and are
faulty. In this paper we describe all traps disconnected halfway with the
size , and discuss the problem whether there exist small sets of
faulty edges which preclude Hamiltonian cycles and are not based on sets
disconnected halfway. We describe heuristic which detects sets of faulty edges
which preclude HC also those sets that are not based on subgraphs disconnected
halfway. We describe all cubes that are not Hamiltonian, and all
cubes with 8 or 9 faulty edges that are not Hamiltonian