2,184 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
How Low Can You Go? New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes
In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The k-planar crossing number of a graph cr k ( G) is the number of crossings required when every edge of G must be drawn in one of k distinct planes. It was shown in [2] that cr 2 ( Q 8 ) ≤ 256 which we improve to cr 2 ( Q 8 ) ≤ 128. Our approach highlights the relationship between symmetric drawings and the study of k-planar crossing numbers. We conclude with several open questions concerning this relationship
Drawing a Graph in a Hypercube
A -dimensional hypercube drawing of a graph represents the vertices by
distinct points in , such that the line-segments representing the
edges do not cross. We study lower and upper bounds on the minimum number of
dimensions in hypercube drawing of a given graph. This parameter turns out to
be related to Sidon sets and antimagic injections.Comment: Submitte
An ETH-Tight Exact Algorithm for Euclidean TSP
We study exact algorithms for {\sc Euclidean TSP} in . In the
early 1990s algorithms with running time were presented for
the planar case, and some years later an algorithm with
running time was presented for any . Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on {\sc Euclidean TSP}, except for a lower bound stating that the
problem admits no algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of {\sc Euclidean
TSP} by giving a algorithm and by showing that a
algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201
Quantum search algorithms on a regular lattice
Quantum algorithms for searching one or more marked items on a d-dimensional
lattice provide an extension of Grover's search algorithm including a spatial
component. We demonstrate that these lattice search algorithms can be viewed in
terms of the level dynamics near an avoided crossing of a one-parameter family
of quantum random walks. We give approximations for both the level-splitting at
the avoided crossing and the effectively two-dimensional subspace of the full
Hilbert space spanning the level crossing. This makes it possible to give the
leading order behaviour for the search time and the localisation probability in
the limit of large lattice size including the leading order coefficients. For
d=2 and d=3, these coefficients are calculated explicitly. Closed form
expressions are given for higher dimensions
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