53,787 research outputs found
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that if is any bridge-addable class of graphs on n vertices, and is taken uniformly at random from , then is connected with probability at least , when n tends to infinity. This lower bound is asymptotically best possible since it is reached for forests. Our proof uses a “local double counting” strategy that may be of independent interest, and that enables us to compare the size of two sets of combinatorial objects by solving a related multivariate optimization problem. In our case, the optimization problem deals with partition functions of trees relative to a supermultiplicative functional.Postprint (author's final draft
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte has described in the book "Connectivity in graphs" a canonical
decomposition of any graph into 3-connected components. In this article we
translate (using the language of symbolic combinatorics)
Tutte's decomposition into a general grammar expressing any family of graphs
(with some stability conditions) in terms of the 3-connected subfamily. A key
ingredient we use is an extension of the so-called dissymmetry theorem, which
yields negative signs in the grammar.
As a main application we recover in a purely combinatorial way the analytic
expression found by Gim\'enez and Noy for the series counting labelled planar
graphs (such an expression is crucial to do asymptotic enumeration and to
obtain limit laws of various parameters on random planar graphs). Besides the
grammar, an important ingredient of our method is a recent bijective
construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
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