8,937 research outputs found
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph contains a given graph as a substructure. When
asking for the graph as a topological minor, i.e., for a copy of a subdivision
of the given graph, it is well-known that the (sharp) threshold is at .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -dimensional complex.Comment: 15 page
Feynman Diagrams and Rooted Maps
The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a
powerful tool for investigating the topological properties of Feynman diagrams,
related to the single particle propagator in the quantum many-body systems. The
numerical correspondence between the number of this class of Feynman diagrams
as a function of perturbative order and the number of rooted maps as a function
of the number of edges is studied. A graphical procedure to associate Feynman
diagrams and rooted maps is then stated. Finally, starting from rooted maps
principles, an original definition of the genus of a Feynman diagram, which
totally differs from the usual one, is given.Comment: 20 pages, 30 figures, 3 table
Fixed parameter tractable algorithms in combinatorial topology
To enumerate 3-manifold triangulations with a given property, one typically
begins with a set of potential face pairing graphs (also known as dual
1-skeletons), and then attempts to flesh each graph out into full
triangulations using an exponential-time enumeration. However, asymptotically
most graphs do not result in any 3-manifold triangulation, which leads to
significant "wasted time" in topological enumeration algorithms. Here we give a
new algorithm to determine whether a given face pairing graph supports any
3-manifold triangulation, and show this to be fixed parameter tractable in the
treewidth of the graph.
We extend this result to a "meta-theorem" by defining a broad class of
properties of triangulations, each with a corresponding fixed parameter
tractable existence algorithm. We explicitly implement this algorithm in the
most generic setting, and we identify heuristics that in practice are seen to
mitigate the large constants that so often occur in parameterised complexity,
highlighting the practicality of our techniques.Comment: 16 pages, 9 figure
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Structure and enumeration of (3+1)-free posets
A poset is (3+1)-free if it does not contain the disjoint union of chains of
length 3 and 1 as an induced subposet. These posets play a central role in the
(3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have
enumerated (3+1)-free posets in the graded case by decomposing them into
bipartite graphs, but until now the general enumeration problem has remained
open. We give a finer decomposition into bipartite graphs which applies to all
(3+1)-free posets and obtain generating functions which count (3+1)-free posets
with labelled or unlabelled vertices. Using this decomposition, we obtain a
decomposition of the automorphism group and asymptotics for the number of
(3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to
clarify the construction of skeleta and the enumeration. An extended abstract
of this paper appears as arXiv:1212.535
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