3,276 research outputs found
The crossing number of locally twisted cubes
The {\it crossing number} of a graph is the minimum number of pairwise
intersections of edges in a drawing of . Motivated by the recent work
[Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper
bound on the crossing number of the hypercube. J. Graph Theory {\bf 59},
145--161 (2008)] which solves the upper bound conjecture on the crossing number
of -dimensional hypercube proposed by Erd\H{o}s and Guy, we give upper and
lower bounds of the crossing number of locally twisted cube, which is one of
variants of hypercube.Comment: 17 pages, 12 figure
The Perfect Local Theorem and Twisted Martingale Transforms
A local Tb Theorem provides a flexible framework for proving the boundedness
of a Calder\'on-Zygmund operator T. One needs only boundedness of the operator
T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes.
We give a new proof of this Theorem in the setting of perfect (dyadic) models
of Calder\'on-Zygmund operators, imposing integrability conditions on the b_Q
functions that are the weakest possible. The proof is a simple direct argument,
based upon a new inequality for transforms of so-called twisted martingale
differences.Comment: 12 pages v2: typos corrected in section 5 v3: The twisted martingale
transform inequalities appeared in Auscher Routin (arXiv:1011.1747), which is
now reflected in the tex
Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes
The -dimensional hypercube network is one of the most popular
interconnection networks since it has simple structure and is easy to
implement. The -dimensional locally twisted cube, denoted by , an
important variation of the hypercube, has the same number of nodes and the same
number of connections per node as . One advantage of is that the
diameter is only about half of the diameter of . Recently, some
interesting properties of were investigated. In this paper, we
construct two edge-disjoint Hamiltonian cycles in the locally twisted cube
, for any integer . The presence of two edge-disjoint
Hamiltonian cycles provides an advantage when implementing algorithms that
require a ring structure by allowing message traffic to be spread evenly across
the locally twisted cube.Comment: 7 pages, 4 figure
Edge-Fault Tolerance of Hypercube-like Networks
This paper considers a kind of generalized measure of fault
tolerance in a hypercube-like graph which contain several well-known
interconnection networks such as hypercubes, varietal hypercubes, twisted
cubes, crossed cubes and M\"obius cubes, and proves for any with by the induction on
and a new technique. This result shows that at least edges of
have to be removed to get a disconnected graph that contains no vertices of
degree less than . Compared with previous results, this result enhances
fault-tolerant ability of the above-mentioned networks theoretically
On Weingarten transformations of hyperbolic nets
Weingarten transformations which, by definition, preserve the asymptotic
lines on smooth surfaces have been studied extensively in classical
differential geometry and also play an important role in connection with the
modern geometric theory of integrable systems. Their natural discrete analogues
have been investigated in great detail in the area of (integrable) discrete
differential geometry and can be traced back at least to the early 1950s. Here,
we propose a canonical analogue of (discrete) Weingarten transformations for
hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and
discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove
the existence of Weingarten pairs and analyse their geometric and algebraic
properties.Comment: 41 pages, 30 figure
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