50,750 research outputs found
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
The enumeration of generalized Tamari intervals
Let be a grid path made of north and east steps. The lattice
, based on all grid paths weakly above and
sharing the same endpoints as , was introduced by Pr\'eville-Ratelle and
Viennot (2014) and corresponds to the usual Tamari lattice in the case
. Our main contribution is that the enumeration of intervals in
, over all of length , is given by . This formula was first obtained by Tutte(1963) for
the enumeration of non-separable planar maps. Moreover, we give an explicit
bijection from these intervals in to non-separable
planar maps.Comment: 19 pages, 11 figures. Title changed, originally titled "From
generalized Tamari intervals to non-separable planar maps (extended
abstract)", submitte
On the effect of projections on convergence peak counts and Minkowski functionals
The act of projecting data sampled on the surface of the celestial sphere
onto a regular grid on the plane can introduce error and a loss of information.
This paper evaluates the effects of different planar projections on
non-Gaussian statistics of weak lensing convergence maps. In particular we
investigate the effect of projection on peak counts and Minkowski Functionals
(MFs) derived from convergence maps and the suitability of a number of
projections at matching the peak counts and MFs obtained from a sphere. We find
that the peak counts derived from planar projections consistently overestimate
the counts at low SNR thresholds and underestimate at high SNR thresholds
across the projections evaluated, although the difference is reduced when
smoothing of the maps is increased. In the case of the Minkowski Functionals,
V0 is minimally affected by projection used, while projected V1 and V2 are
consistently overestimated with respect to the spherical case
Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof
We consider the problem of enumeration of planar maps and revisit its
one-matrix model solution in the light of recent combinatorial techniques
involving conjugated trees. We adapt and generalize these techniques so as to
give an alternative and purely combinatorial solution to the problem of
counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Basic properties of the infinite critical-FK random map
We investigate the critical Fortuin-Kasteleyn (cFK) random map model. For
each and integer , this model chooses a planar map
of edges with a probability proportional to the partition function of
critical -Potts model on that map. Sheffield introduced the
hamburger-cheeseburger bijection which maps the cFK random maps to a family of
random words, and remarked that one can construct infinite cFK random maps
using this bijection. We make this idea precise by a detailed proof of the
local convergence. When , this provides an alternative construction of the
UIPQ. In addition, we show that the limit is almost surely one-ended and
recurrent for the simple random walk for any , and mutually singular in
distribution for different values of .Comment: 14 pages, 6 figures. v2: Fixed the proof of main theorem, removed old
lemma 5, added results on mutually singular measures and ergodicity.
Submitted to Annales de l'Institut Henri Poincar\'e
Semi-Global Stereo Matching with Surface Orientation Priors
Semi-Global Matching (SGM) is a widely-used efficient stereo matching
technique. It works well for textured scenes, but fails on untextured slanted
surfaces due to its fronto-parallel smoothness assumption. To remedy this
problem, we propose a simple extension, termed SGM-P, to utilize precomputed
surface orientation priors. Such priors favor different surface slants in
different 2D image regions or 3D scene regions and can be derived in various
ways. In this paper we evaluate plane orientation priors derived from stereo
matching at a coarser resolution and show that such priors can yield
significant performance gains for difficult weakly-textured scenes. We also
explore surface normal priors derived from Manhattan-world assumptions, and we
analyze the potential performance gains using oracle priors derived from
ground-truth data. SGM-P only adds a minor computational overhead to SGM and is
an attractive alternative to more complex methods employing higher-order
smoothness terms.Comment: extended draft of 3DV 2017 (spotlight) pape
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