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.
We introduce a new cohomology theory for planar trivalent graphs with perfect matchings. The graded Euler characteristic of the cohomology is a one variable polynomial called the 2-factor polynomial that, if nonzero when evaluated at one, implies that the perfect matching is even and therefore the graph is 4-face colorable. We also define several new polynomials invariants of graphs with and without perfect matchings that are invariants of abstract tensors systems and spin networks defined by Roger Penrose in the 1970s. We show how some of these polynomials can be ``categorified'' into their o…
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.
The focus of this paper is on discussion of a catalog of a class of (3, g) graphs for even girth g. A (k, g) graph is a graph with regular degree k and girth g. This catalog is compared with other known lists of (3, g) graphs such as the enumerations of trivalent symmetric graphs and enumerations of trivalent vertex-transitive graphs, to conclude that this catalog has graphs for more orders than these lists. This catalag also specifies a list of orders, rotational symmetry and girth for which the class of (3, g) graphs do not exist. It is also shown that this catalog of graphs extends infinite…
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.
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.
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.
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…
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(…
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 study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus $7$. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically $0\%$ of connected trivalent planar graphs are tropically planar.
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…
$β(1,0)$-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.
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-…
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.
Let $v$ be a grid path made of north and east steps. The lattice $\rm{T{\scriptsize AM}}(v)$, based on all grid paths weakly above $v$ and sharing the same endpoints as $v$, was introduced by Préville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case $v=(NE)^n$. Our main contribution is that the enumeration of intervals in $\rm{T{\scriptsize AM}}(v)$, over all $v$ of length $n$, is given by $\frac{2 (3n+3)!}{(n+2)! (2n+3)!}$. This formula was first obtained by Tutte(1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection …
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…
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…
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…
Enumeration of general planar hypermaps with an alternating boundary
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；CEA（法国原子能和替代能源委员会）；Institut de physique théorique（理论物理研究所）；Institut de Mathématiques d’Orsay（奥赛数学研究所）；Fakultät für Mathematik, Universität Wien（维也纳大学数学系）；CRM, Centre de recherches mathématiques de Montréal（蒙特利尔数学研究中心）
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…
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.
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…
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…
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.
In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.
In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighbourhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straight…
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 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…
In 2006, Chapoton defined a class of Tamari intervals called "new intervals" in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called "degree tree". Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.
We present recent results on the enumeration of $q$-coloured planar maps, where each monochromatic edge carries a weight $ν$. This is equivalent to weighting each map by its Tutte polynomial, or to solving the $q$-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (non-linear) differential equation. The starting point of this result is a functional equation written by Tutte in 1971, which translates into enumerative terms a simple recursive description of planar maps. The…
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.
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 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…
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.
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.
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 extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels.…
