733 research outputs found
Geometry of tropical moduli spaces and linkage of graphs
We prove the following "linkage" theorem: two p-regular graphs of the same
genus can be obtained from one another by a finite alternating sequence of
one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the
linkage theorem to prove that various moduli spaces of tropical curves are
connected through codimension one.Comment: Final version incorporating the referees correction
Stronger ILPs for the Graph Genus Problem
The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice.
Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding.
Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Topological Quantum Error Correction with Optimal Encoding Rate
We prove the existence of topological quantum error correcting codes with
encoding rates asymptotically approaching the maximum possible value.
Explicit constructions of these topological codes are presented using surfaces
of arbitrary genus. We find a class of regular toric codes that are optimal.
For physical implementations, we present planar topological codes.Comment: REVTEX4 file, 5 figure
Hamiltonian cycles and 1-factors in 5-regular graphs
It is proven that for any integer and ,
there exist infinitely many 5-regular graphs of genus containing a
1-factorisation with exactly pairs of 1-factors that are perfect, i.e. form
a hamiltonian cycle. For , this settles a problem of Kotzig from 1964.
Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing
techniques aimed at producing graphs of high cyclic edge-connectivity. We prove
that there exist infinitely many planar 5-connected 5-regular graphs in which
every 1-factorisation has zero perfect pairs. On the other hand, by the Four
Colour Theorem and a result of Brinkmann and the first author, every planar
4-connected 5-regular graph satisfying a condition on its hamiltonian cycles
has a linear number of 1-factorisations each containing at least one perfect
pair. We also prove that every planar 5-connected 5-regular graph satisfying a
stronger condition contains a 1-factorisation with at most nine perfect pairs,
whence, every such graph admitting a 1-factorisation with ten perfect pairs has
at least two edge-Kempe equivalence classes. The paper concludes with further
results on edge-Kempe equivalence classes in planar 5-regular graphs.Comment: 27 pages, 13 figures; corrected figure
(3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
We apply the recently suggested strategy to lift state spaces and operators
for (2+1)-dimensional topological quantum field theories to state spaces and
operators for a (3+1)-dimensional TQFT with defects. We start from the
(2+1)-dimensional Turaev-Viro theory and obtain a state space, consistent with
the state space expected from the Crane-Yetter model with line defects. This
work has important applications for quantum gravity as well as the theory of
topological phases in (3+1) dimensions. It provides a self-dual quantum
geometry realization based on a vacuum state peaked on a homogeneously curved
geometry. The state spaces and operators we construct here provide also an
improved version of the Walker-Wang model, and simplify its analysis
considerably. We in particular show that the fusion bases of the
(2+1)-dimensional theory lead to a rich set of bases for the (3+1)-dimensional
theory. This includes a quantum deformed spin network basis, which in a loop
quantum gravity context diagonalizes spatial geometry operators. We also obtain
a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.
Furthermore, the construction presented here can be generalized to provide
state spaces for the recently introduced dichromatic four-dimensional manifold
invariants.Comment: 27 pages, many figures, v2: minor correction
Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
We give a classification of generic coadjoint orbits for the groups of
symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic
surface. We also classify simple Morse functions on symplectic surfaces with
respect to actions of those groups. This gives an answer to V.Arnold's problem
on describing all invariants of generic isovorticed fields for the 2D ideal
fluids. For this we introduce a notion of anti-derivatives on a measured Reeb
graph and describe their properties.Comment: 38 pages, 11 figures; to appear in Annales de l'Institut Fourie
Birational cobordism invariance of uniruled symplectic manifolds
A symplectic manifold is called {\em (symplectically) uniruled}
if there is a nonzero genus zero GW invariant involving a point constraint. We
prove that symplectic uniruledness is invariant under symplectic blow-up and
blow-down. This theorem follows from a general Relative/Absolute correspondence
for a symplectic manifold together with a symplectic submanifold. A direct
consequence is that symplectic uniruledness is a symplectic birational
invariant. Here we use Guillemin and Sternberg's notion of cobordism as the
symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat
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