8,154 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
Properly ordered dimers, -charges, and an efficient inverse algorithm
The superconformal field theories that arise in AdS-CFT from
placing a stack of D3-branes at the singularity of a toric Calabi-Yau threefold
can be described succinctly by dimer models. We present an efficient algorithm
for constructing a dimer model from the geometry of the Calabi-Yau. Since not
all dimers produce consistent field theories, we perform several consistency
checks on the field theories produced by our algorithm: they have the correct
number of gauge groups, their cubic anomalies agree with the Chern-Simons
coefficients in the AdS dual, and all gauge invariant chiral operators satisfy
the unitarity bound. We also give bounds on the ratio of the central charge of
the theory to the area of the toric diagram. To prove these results, we
introduce the concept of a properly ordered dimer.Comment: 33 pages, 19 figures, some corrections and clarification
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