9,265 research outputs found
A bijection for plane graphs and its applications
International audienceThis paper is concerned with the counting and random sampling of plane graphs (simple planar graphs embedded in the plane). Our main result is a bijection between the class of plane graphs with triangular outer face, and a class of oriented binary trees. The number of edges and vertices of the plane graph can be tracked through the bijection. Consequently, we obtain counting formulas and an efficient random sampling algorithm for rooted plane graphs (with arbitrary outer face) according to the number of edges and vertices. We also obtain a bijective link, via a bijection of Bona, between rooted plane graphs and 1342-avoiding permutations. 1 Introduction A planar graph is a graph that can be embedded in the plane (drawn in the plane without edge crossing). A pla-nar map is an embedding of a connected planar graph considered up to deformation. The enumeration of pla-nar maps has been the subject of intense study since the seminal work of Tutte in the 60's [20] showing that many families of planar maps have beautiful counting formulas. Starting with the work of Cori and Vauquelin [10] and then Schaeffer [18, 19], bijective constructions have been discovered that provide more transparent proofs of such formulas. The enumeration of planar graphs has also been the focus of a lot of efforts, culminating with the asymptotic counting formulas obtained by Giménez and Noy [16]. In this paper we focus on simple planar maps (planar maps without loops nor multiple edges), which are also called plane graphs. This family of planar maps has, quite surprisingly, not been considered until fairly recently. This is probably due to the fact that loops and multiple edges are typically allowed in studies about planar maps, whereas they are usually forbidden in studies about planar graphs. At any rate, the first result about plane graphs was an exact algebraic expressio
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root
Transversal structures on triangulations: a combinatorial study and straight-line drawings
This article focuses on a combinatorial structure specific to triangulated
plane graphs with quadrangular outer face and no separating triangle, which are
called irreducible triangulations. The structure has been introduced by Xin He
under the name of regular edge-labelling and consists of two bipolar
orientations that are transversal. For this reason, the terminology used here
is that of transversal structures. The main results obtained in the article are
a bijection between irreducible triangulations and ternary trees, and a
straight-line drawing algorithm for irreducible triangulations. For a random
irreducible triangulation with vertices, the grid size of the drawing is
asymptotically with high probability up to an additive
error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for
these triangulations only guarantees a grid size .Comment: 42 pages, the second version is shorter, focusing on the bijection
(with application to counting) and on the graph drawing algorithm. The title
has been slightly change
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
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