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Counting rooted near-triangulations on the sphere

Fengming Dong, Yanpei Liu
1993 Discrete Mathematics  
This paper provides the results on the enumerations of rooted simple outerplanar maps, rooted outerplanar near-triangulations, rooted 2-connected near-triangulations, rooted strict 2-connected near-triangulations  ...  and rooted simple 2-connected near-triangulations.  ...  f=O-82 We now construct a function g(x,z) as follows: d--O2 x= 1 +z02(1 -0)' g=o-82. (6.6) (6.7) (6.8) Counting rooted near-triangulations OH the sphere It can be seen that g(x, 1) =f(x).  ... 
doi:10.1016/0012-365x(93)90005-e fatcat:lgfmgfsujbd5dpsxkqybedpsxi

Submaps of maps. III. k-Connected nonplanar maps

Edward A Bender, L.Bruce Richmond
1992 Journal of combinatorial theory. Series B (Print)  
We study 3-connected maps, 3-connected triangulations. and simple triangulations on arbitrary surfaces.  ...  It is shown that the radii of convergence do not depend on the surface and that the number of maps grows in a smooth manner.  ...  a canonical way in the root face of a map counted by m,(O).  ... 
doi:10.1016/0095-8956(92)90036-w fatcat:lxawkh3agfhtnl3cv5hcnybg64

RANDOM PLANAR GRAPHS AND THE NUMBER OF PLANAR GRAPHS [chapter]

Marc Noy
2007 Combinatorics, Complexity, and Chance  
We survey recent progress on the enumeration of labelled planar graph and on the distribution of several parameters in random planar graphs.  ...  The emphasis is on presenting the main ideas while keeping technical details to a minimum.  ...  Let us sketch how Tutte [21] solved the problem of counting near-triangulations.  ... 
doi:10.1093/acprof:oso/9780198571278.003.0014 fatcat:o2zbivji25gglf5yt2psdsxvri

Page 5579 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
Haruo Hosoya (J-OCHS-I; Bunkyo) 94j:05064 05C30 05A15 Dong, Feng Ming (PRC-ASBJ-AM; Beijing); Liu, Yan Pei (PRC-ASBJ-AM; Beijing) Counting rooted near-triangulations on the sphere.  ...  The authors obtain formulas for the number of the following rooted maps enumerated by the num- ber of edges and root face degree: simple outerplanar, outerplanar near-triangulations, 2-connected near-triangulations  ... 

Page 5319 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
Summary: “It is shown that constructing generating functions with one even and one odd variable can help in the weighted counting of planar rooted trees.  ...  A near-triangulation is of type [n,m] if it has n interior vertices, and has m vertices in the boundary of the exterior face; it is rooted if an edge of the exterior face is selected as the root, and oriented  ... 

A survey of the asymptotic behaviour of maps

Edward A Bender, L.Bruce Richmond
1986 Journal of combinatorial theory. Series B (Print)  
A survey is given of the asymptotic enumeration of maps. The asymptotic formulas for both rooted and unrooted maps on surfaces are discussed.  ...  The techniques used to derive these results are briefly discribed. The survey concludes with open problems and areas for future research.  ...  The Note added in proof: The conjectured asymptotic formula for the number of maps on a fixed surface has been verified for l-c maps on any surface by Bender and Canlield and for 2-c maps by Bender and  ... 
doi:10.1016/0095-8956(86)90086-9 fatcat:k5lrnkgi55hirgoeifwhpgkkqe

Uniform Infinite Planar Triangulation and Related Time-Reversed Critical Branching Process

M. A. Krikun
2005 Journal of Mathematical Sciences  
We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process.  ...  This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity).  ...  Multi-rooted triangulations One can consider RNT as a triangulation of a disk, or of a sphere with a hole. The disk is obtained from a sphere by cutting the rooted face of a RNT.  ... 
doi:10.1007/s10958-005-0424-4 fatcat:3qkyn2rqq5dv5dlwmkl7dsaqne

The number of rooted 2-connected triangular maps on the projective plane

Zhi-Cheng Gao
1991 Journal of combinatorial theory. Series B (Print)  
Let s0 and pii be the number of rooted loopless near-triangular maps on the sphere and projective plane, respectively, having i vertices and root face valency j.  ...  Since a single vertex is a map only on the sphere, which is counted by the last term 1 in (2.1), we will assume that our map has edges from now on.  ... 
doi:10.1016/0095-8956(91)90058-r fatcat:hpcdg2s7qjeqfcvc23pdn7cjwe

Page 4231 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
near-triangulations on the Mobius band.  ...  In this paper, we count rooted non- separable near-triangulations on the Mdbius band by the root- face valency, the size and the number of inner faces, and give a parametric expression by which an explicit  ... 

Counting planar maps, coloured or uncoloured [article]

Mireille Bousquet-Mélou
2020 arXiv   pre-print
The proof follows and adapts Tutte's solution of properly q-coloured triangulations (1973-1984).  ...  We present recent results on the enumeration of q-coloured planar maps, where each monochromatic edge carries a weight ν.  ...  Acknowledgements The author gratefully acknowledges the assistance of Olivier Bernardi in writing this survey.  ... 
arXiv:2004.08792v1 fatcat:3yj3frugm5e7zp6tkxuxbyot7m

The Geometry of the Elongated Phase in 4-D Simplicial Quantum Gravity [article]

Gabriele Gionti
1997 arXiv   pre-print
In particular using Walkup's theorem we prove that the dominating configurations in the elongated phase are tree-like structures called "stacked spheres".  ...  Such configurations can be mapped into branched polymers and baby universes arguments are used in order to analyse the critical behaviour of theory in the weak coupling regime.  ...  an upper bound on the number of , respectively, rooted and unrooted inequivalent stacked spheres.  ... 
arXiv:hep-lat/9711018v2 fatcat:6ntoobaiqvg6rmymkessauxzpq

Floer theory of higher rank quiver 3-folds [article]

Ivan Smith
2020 arXiv   pre-print
An ideal triangulation of S defines, for each rank m, a quiver Q(Δ_m), hence a CY_3-category (C,W) for any potential W on Q(Δ_m).  ...  We show that for ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of (Y,ω) is quasi-isomorphic to (C,W_[ω]) for a certain generic potential W_[ω].  ...  Definition 4.5 An eigen-ordering of a generic tuple Φ is a choice of ordering of the roots of Φ(b) = 0 near each point p ∈ D.  ... 
arXiv:2002.10735v1 fatcat:awhzmyik6nd6lgq7fbcyeydq7y

On Triangulations with High Vertex Degree

Olivier Bernardi
2008 Annals of Combinatorics  
The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces.  ...  Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration  ...  Consider a non-separable near-triangulation distinct from L rooted on a digon (i.e. the root-face has degree 2).  ... 
doi:10.1007/s00026-008-0334-5 fatcat:iol5tiajibciheaxlatretspym

On triangulations with high vertex degree [article]

Olivier Bernardi
2006 arXiv   pre-print
The originality of the problem lies in the fact that degree restrictions are placed both on vertices and faces.  ...  Our proofs first follow Tutte's classical approach: we decompose maps by deleting the root and translate the decomposition into an equation satisfied by the generating function of the maps under consideration  ...  Consider a non-separable near-triangulation distinct from L rooted on a digon (i.e. the root-face has degree 2).  ... 
arXiv:math/0601678v1 fatcat:5izt573bendurkdiyeye2ocz7u

On the shape of a curve

Raoul Bott
1975 Advances in Mathematics  
Near the branch points (the cube roots of 1) one argues as follows. Let p be such a cube root so that p3 = 1.  ...  Now since our branch points are vertexes in the triangulation, it is easy to see that the inverse image of the triangulation under r is a triangulation of S and simply counting we see that the number d  ... 
doi:10.1016/0001-8708(75)90147-4 fatcat:aw7e4oklyrhfrpdsz2y6etyyxu
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