Omer Gimenez, Marc Noy
We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.
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We consider planar cubic maps, i.e. connected cubic graphs imbedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is $k$-th Catalan number.
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes.
We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.
We show that the number $g_n$ of labelled series-parallel graphs on $n$ vertices is asymptotically $g_n \sim g\cdot n^{-5/2} γ^n n!$, where $γ$ and $g$ are explicit computable constants. We show that the number of edges in random series-parallel graphs is asymptotically normal with linear mean and variance, and that the number of edges is sharply concentrated around its expected value. Similar results are proved for labelled outerplanar graphs and for graphs not containing $K_{2,3}$ as a minor.
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.
This work is a follow-up of the article [Proc.\ London Math.\ Soc.\ 119(2):358--378, 2019], where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$ vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} ρ^{-n} n!$, where $g>0$ is a constant and $ρ\approx 0.24377$ is the radius of convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of several classes of…
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give addition…
Asymptotic enumeration of unlabelled cubic planar graphs
本文结合生成函数方法、计算界和概率局部大偏差定理，精确确定了n个顶点的无标号三次平面图的渐近数量。
Enumeration of rooted 3-connected bipartite planar maps
本研究首次解决有根3-连通二部平面图的计数问题，通过分解图的连通分量得到对应生成函数，推导出该类图数量的渐近估计式。
Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating function $H(x)$ of all 1342-avoiding permutations of length $n$ as well as an {\em exact} formula for their number $S_n(1342)$. While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of length $n$ equals that of labeled plane trees of a certain type on $n$ vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(…
It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}γ^n n!$ where $c^{(g)}>0$, and $γ\approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-…
In this paper we construct inverse bijections between two sequences of finite sets. One sequence is defined by planar diagrams and the other by lattice walks. G. Kuperberg has shown that the number of elements in these two sets are equal. This problem and the methods we use are motivated by the representation theory of the exceptional simple Lie algebra $G_2$. However in this account we have emphasised the combinatorics.
Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks
本文研究了水平2谱系网络子类的渐近计数，揭示了终端平面网络对增长率的显著影响及不同子类间的增长速率差异。
Enumeration of planar bipartite tight irreducible maps
本文研究平面二部紧且2b-不可约地图的计数问题，通过基于切片分解的直接双射方法，将地图分解为2b-不可约紧切片和双面地图，并利用b-装饰树进行双射编码，最终给出了该类地图数量的显式多项式表达式。
$β(1,0)$-trees provide a convenient description of rooted non-separable planar maps. The involution $h$ on $β(1,0)$-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In this paper, we describe and enumerate fixed points of the involution $h$. Intriguingly, the fixed points are equinumerous with the fixed points under taking the dual map on rooted non-separable planar maps, even though the fixed points do not go to each other under the know (natural) bijection between the trees and the maps.
Lattice paths effectively model phenomena in chemistry, physics and probability theory. Asymptotic enumeration of lattice paths is linked with entropy in the physical systems being modeled. Lattice paths restricted to different regions of the plane are well suited to a functional equation approach for exact and asymptotic enumeration. This thesis surveys results on lattice paths under various restrictions, with an emphasis on lattice paths in the quarter plane. For these paths, we develop an original systematic combinatorial approach providing direct access to the exponential growth factors of…
We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} γ^n n!$ for a constant $c_1>0$ and $γ\approx 11.89235$. We also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} δ^n$, where $δ= 1/σ\approx 6.40375$, and $σ$ is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively r…
We consider the class F of 2-connected non-planar K_{3,3}-subdivision-free graphs that are embeddable in the projective plane. We show that these graphs admit a unique decomposition as a graph K_5 (the core) where the edges are replaced by two-pole networks constructed from 2-connected planar graphs. A method to enumerate these graphs in the labelled case is described. Moreover, we enumerate the homeomorphically irreducible graphs in F and homeomorphically irreducible 2-connected planar graphs. Particular use is made of two-pole directed series-parallel networks. We also show that the number m…
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type p…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
We split the crossings of the foil knot and enumerate the resulting states with a generating polynomial. Unexpectedly, the number of such states which consist of two components are given by the lazy caterer's sequence. This sequence describes the maximum number of planar regions that is obtained with a given number of straight lines. We then establish a bijection between this partition of the plane and the concerned foil splits sequence.
Refined enumeration of planar Eulerian orientations
通过组合论证和函数方程，研究四价平面地图欧拉定向按顶点数、交替顶点数和顺时针面数的三参数精细枚举，并求解三种特殊情况。
We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.
We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with out-degrees prescribed by a function $å:V\to \NN$ unifies many different combinatorial structures, including the afore mentioned. We call these orientations $å$-orientations. The main focus of this paper are bounds for the maximum number of $å$-orientations that a planar map with $n$ vertices can have, for different instances of $å$. We give examples of triangu…
Enumeration formulas for (3, 6)-fullerenes
针对所有面为三角形或六边形的三次平面图(3,6)-富勒烯，给出了顶点数为V的精确计数公式，并进一步枚举了具有镜像对称、三重旋转对称及两种对称性的情况，公式以V的素因子分解表示。
The generating function of planar Eulerian orientations
该研究通过将生成函数表示为简单超几何级数的反函数，解决了平面欧拉定向及4价平面欧拉定向的计数问题，验证了此前的渐近行为预测，并发现了4价情形下的意外组合关联。
Efficient Enumeration of Drawings and Combinatorial Structures for Maximal Planar Graphs
本文通过枚举极大平面图的规范定向，并借助其与规范序、Schnyder 木及两类经典图画的双射关系，实现了 $O(n)$ 时间、空间和延迟的高效枚举算法。
A connected planar cubic graph is called an $m$-barrel fullerene and denoted by $F(m,k)$, if it has the following structure: The first circle is an $m$-gon. Then $m$-gon is bounded by $m$ pentagons. After that we have additional k layers of hexagons. At the last circle $m$-pentagons connected to the second $m$-gon. In this paper we asymptotically count by two different methods the number of perfect matchings in $m$-barrel fullerene graphs, as the number of hexagonal layers is large, and show that the results are equal.
We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers $D$. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.
Given a sequence of integers $a_j, j\ge 1,$ a multiset is a combinatorial object composed of unordered components, such that there are exactly $a_j$ one-component multisets of size $j.$ When $a_j\asymp j^{r-1} y^j$ for some $r>0$, $y\geq 1$, then the multiset is called {\em expansive}. Let $c_n$ be the number of multisets of total size $n$. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that $c_n/c_{n+1}<1$ for large enough $n$. This allows us to prove Monadic Second Order Limit Laws for expansive multisets. The above results are extended to a clas…
A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of innerfaces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations - hypermaps whose all…
Consider a family $\mathcal{T}$ of 3-connected graphs of moderate growth, and let $\mathcal{G}$ be the class of graphs whose 3-connected components are graphs in $\mathcal{T}$. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and series-parallel graphs. We provide a general result for the asymptotic number of graphs in $\mathcal{G}$, based on the singularities of the exponential generating function associated to $\mathcal{T}$. We derive limit laws, which are e…
We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working…
We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and havin…
Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that the distance between any two points lies in $\{1,..., r\}$. Our main result describes their approximate structure for large $n$. As a consequence, we show that the number of these metric spaces is $\lceil \frac{r+1}{2}\rceil ^{{n\choose 2} + o(n^2)}$. Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij. When $r$ is even, our structural characterization is more precise, and implies that almost all such metric spaces have all distances at least $r/2$. As an easy c…
Enumeration of general planar hypermaps with an alternating boundary
本文扩展了之前工作中关于具有交替边界的平面超图的枚举研究，通过经典的一对多对应关系，提出了一种新的策略来获得一般情况下的代数方程，包括装饰有伊辛模型的图，并通过同时消除两个催化变量的主要步骤，展示了伊辛四边形化情况下的显式有理参数化，从而证明了一些显著性质在一般情况下不再成立。
Double-scaled SYK from boundary metrics of planar maps
本文研究一类具有特殊q变形面权重的二分平面地图模型，证明其边界度量枚举仅依赖于测地线弦图的交叉数，并与双尺度SYK模型的弦图表示一致。
In this paper, we generalize 2-trees by replacing triangles by quadrilaterals, pentagons or $k$-sided polygons ($k$-gons), where $k\geq 3$ is fixed. This generalization, to $k$-gonal 2-trees, is natural and is closely related, in the planar case, to some specializations of the cell-growth problem. Our goal is the labelled and unlabelled enumeration of $k$-gonal 2-trees according to the number $n$ of $k$-gons. We give explicit formulas in the labelled case, and, in the unlabelled case, recursive and asymptotic formulas.
Counting colored planar maps: algebraicity results
本文证明当 q 为特定形式（含 q=2,3）时，适当 q 着色平面地图及 Potts 加权地图的生成函数是代数的，并推广至三角剖分，首次求解了带两个催化变量的非线性方程。
The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher's result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumer…
For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects paired with their multiplicities. For example, $\mathrm{MSET}(\mathbb{N})$ is (isomorphic to) the class of number partitions of positive integers, a prominent and well-studied case. The multiset construction appears naturally in the study of unlabelled objects, for example graphs or various structures related to number partitions. Our main result establishes…
We provide precise asymptotic estimates for the number of several classes of labelled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky et al. (Random Structures Algorithms 2007). We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptoticall…
We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.
This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from brute force enumeration, up to the equivalence to a dedicated problem in graph theory. This leads to algorithms starting from an untractable case on modern computers, up to a run of few seconds on a portable personal computer. This emphasizes the need for a prior analysis by humans before considering the assistance of computers for complex design problems. The…
Eulerian orientations and the six-vertex model on planar map
本文用两种方法计数带欧拉定向的平面四价地图，并证明基于Kostov工作的六顶点模型加权计数结果推广了第一种方法得到的两个计数结果。
Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with the same number of leaves and uniquely matching each leaf in $T$ with a leaf in $S$. They are usually represented using layouts, which embed the trees and the matching of the leaves into the plane as in Figure 1. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. …
On the number of planar Eulerian orientations
本文通过构造并计数收敛于平面欧拉定向集合的子集族与超集族，证明其生成函数均为代数函数，从而得到平面欧拉定向增长速率约在12.5附近的上下界。
Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…
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.
Symmetries of (3, 6)-Fullerenes
研究(3, 6)-富勒烯的对称性，通过精确枚举具有特定对称类型的(3, 6)-富勒烯，并结合构造方法，解决了关于给定顶点数的特定对称性(3, 6)-富勒烯存在性的三个猜想。
We prove that for each $k\ge0$, the probability that a root vertex in a random planar graph has degree $k$ tends to a computable constant $d_k$, so that the expected number of vertices of degree $k$ is asymptotically $d_k n$, and moreover that $\sum_k d_k =1$. The proof uses the tools developed by Gimenez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain e…
We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projective-planar graphs containing no K33-subdivisions, we apply these techniques to obtain their unlabelled enumeration.
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for generating functions as…
Functional equations with one catalytic appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions the dominant singularity of the solutions have a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we - usually - have a singularity of type 3/2.
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit probability distributions present in large random structures. ``Singularity analysis'' reviewed here provides constructive estimates that are applicable in several areas of combinatorics. It constitutes a complex-analytic Tauberian procedure by which combinatorial constructions and asymptotic--probabilistic laws can be systematically related.
Catalytic equations appear in several combinatorial applications, most notably in the numeration of lattice path and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size $n$ in various planar map classes grows asymptotically like $c\cdot n^{-5/2} γ^n$, for suitable positive constants $c$ and $γ$. Essentially we have to distinguish between linear catalyti…
We consider ensembles of planar maps with two marked vertices at distance k from each other and look at the closed line separating these vertices and lying at distance d from the first one (d<k). This line divides the map into two components, the hull at distance d which corresponds to the part of the map lying on the same side as the first vertex and its complementary. The number of faces within the hull is called the hull volume and the length of the separating line the hull perimeter. We study the statistics of the hull volume and perimeter for arbitrary d and k in the limit of infinitely l…
Enumeration of maps with the Dumitriu-Edelman model
利用Dumitriu-Edelman三对角模型，给出了β系综粒子幂和累积量在1/N和β下的展开，系数由Bouttier、Fusy和Guitter引入的适当标记地图表示，仅涉及可定向地图，并通过新映射将前两阶与LaCroix的展开联系起来。
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