1,070 research outputs found
A Case Study in Dimensional Deconstruction
We test Arkani-Hamed et al.'s dimensional deconstruction on a model that is
predicted to have a naturally light composite Higgs boson, i.e., one whose mass
M is much less than its binding scale \Lambda, and whose quartic coupling
\lambda is large, so that its vacuum expectation value v \sim M/\sqrt{\lambda}
<< \Lambda also. We consider two different underlying dynamics--UV
completions--at the scale \Lambda for this model. We find that the expectation
from dimensional deconstruction is not realized and that low energy details
depend crucially on the UV completion. In one case, M << \Lambda and \lambda <<
1, hence, v \sim \Lambda. In the other, \lambda can be large or small, but then
so is M, and v is still O(\Lambda).Comment: 20 pages, LaTeX, with 8 postscript figure
Hamiltonian Cycles in Polyhedral Maps
We present a necessary and sufficient condition for existence of a
contractible, non-separating and noncontractible separating Hamiltonian cycle
in the edge graph of polyhedral maps on surfaces. In particular, we show the
existence of contractible Hamiltonian cycle in equivelar triangulated maps. We
also present an algorithm to construct such cycles whenever it exists.Comment: 14 page
The PC-Tree algorithm, Kuratowski subdivisions, and the torus.
The PC-Tree algorithm of Shih and Hsu (1999) is a practical linear-time planarity algorithm that provides a plane embedding of the given graph if it is planar and a Kuratowski subdivision otherwise. Remarkably, there is no known linear-time algorithm for embedding graphs on the torus. We extend the PC-Tree algorithm to a practical, linear-time toroidality test for K3;3-free graphs called the PCK-Tree algorithm. We also prove that it is NP-complete to decide whether the edges of a graph can be covered with two Kuratowski subdivisions. This greatly reduces the possibility of a polynomial-time toroidality testing algorithm based solely on edge-coverings by subdivisions of Kuratowski subgraphs
Hexagonal Tilings: Tutte Uniqueness
We develop the necessary machinery in order to prove that hexagonal tilings
are uniquely determined by their Tutte polynomial, showing as an example how to
apply this technique to the toroidal hexagonal tiling.Comment: 12 figure
Two Results in Drawing Graphs on Surfaces
In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change
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