6 research outputs found
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
The -extra edge-connectivity of balanced hypercubes
The -extra edge-connectivity is an important measure for the reliability
of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320
(2018) 464--473] determined the -extra edge-connectivity of balanced
hypercubes and conjectured that the -extra edge-connectivity of
is for . In this paper,
we confirm their conjecture for and ,
and disprove their conjecture for and , where .Comment: 12 pages, 2 figure
Fault-tolerance of balanced hypercubes with faulty vertices and faulty edges
Let (resp. ) be the set of faulty vertices (resp. faulty edges)
in the -dimensional balanced hypercube . Fault-tolerant Hamiltonian
laceability in with at most faulty edges is obtained in [Inform.
Sci. 300 (2015) 20--27]. The existence of edge-Hamiltonian cycles in
for are gotten in [Appl. Math. Comput. 244 (2014) 447--456].
Up to now, almost all results about fault-tolerance in with only faulty
vertices or only faulty edges. In this paper, we consider fault-tolerant cycle
embedding of with both faulty vertices and faulty edges, and prove that
there exists a fault-free cycle of length in with
and for . Since is a
bipartite graph with two partite sets of equal size, the cycle of a length
is the longest in the worst-case.Comment: 17 pages, 5 figures, 1 tabl
Unpaired many-to-many disjoint path cover of balanced hypercubes
The balanced hypercube , a variant of the hypercube, was proposed as a
desired interconnection network topology. It is known that is bipartite.
Assume that and
are any two sets of vertices in different partite sets of (). It
has been proved that there exists paired 2-disjoint path cover of . In
this paper, we prove that there exists unpaired -disjoint path cover of
() from to , which improved some known results. The upper
bound of the number of disjoint paths in unpaired -disjoint path
cover is best possible.Comment: arXiv admin note: text overlap with arXiv:1804.0194
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