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 triangulations with $2.37^n$ Schnyder woods, 3-connected planar maps with $3.209^n$ Schnyder woods and inner triangulations with $2.91^n$ bipolar orientations. These lower bounds are accompanied by upper bounds of $3.56^n$, $8^n$ and $3.97^n$ respectively. We also show that for any planar map $M$ and any $α$ the number of $α$-orientations is bounded from above by $3.73^n$ and describe a family of maps which have at least $2.598^n$ $α$-orientations.
A graph is (m,k)-colourable if its vertices can be coloured with m colours such that the maximum degree of the subgraph induced on the set of all vertices receiving the same colour is at most k. The k-defective chromatic number $χ_k(G)$ is the least positive integer $m$ for which graph G is (m,k)-colourable. Let f(m,k;tfp) be the smallest order of a triangle-free planar graph such that $χ_k(G)$=m. In this paper we show that f(3,1;tfp)=11.
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.
Let $G$ be a graph embedded on an orientable surface. Given a class ${\cal C}$ of facial circuits of $G$ as a forbidden class, we give a sufficient-necessary condition for that an $α$-orientation (orientation with prescribed out-degrees) of $G$ can be transformed into another by a sequence of flips on non-forbidden circuits and further give an explicit formula for the minimum number of such flips. We also consider the connection among all $α$-orientations by defining a directed graph ${\bf D}({\cal C})$, namely the ${\cal C}$-forbidden flip graph. We show that if ${\cal C}\not=\emptyset$, then…
We show that a graph has an orientation under which every circuit of even length is clockwise odd if and only if the graph contains no subgraph which is, after the contraction of at most one circuit of odd length, an even subdivision of K_{2,3}. In fact we give a more general characterisation of graphs that have an orientation under which every even circuit has a prescribed clockwise parity. This problem was motivated by the study of Pfaffian graphs, which are the graphs that have an orientation under which every alternating circuit is clockwise odd. Their significance is that they are precise…
An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrowχ(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper orientations of $G$. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientatio…
Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coeffi…
This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific "transversal structures" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are re…
Orientations of $10$-Edge-Connected Planar Multigraphs and Applications
William & Mary, Department of Mathematics（威廉与玛丽学院）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）；Henan University of Engineering（河南工程学院）
A bijection $Φ$ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to R. Baxter for the number $Θ_{ij}$ of plane bipolar orientations with $i$ non-polar vertices and $j$ inner faces: $Θ_{ij}=2\frac{(i+j)!(i+j+1)!(i+j+2)!}{i!(i+1)!(i+2)!j!(j+1)!(j+2)!}$. In addition, it is shown that $Φ$ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words.
The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,...,P_n$ are (co…
We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees, with interesting consequences for enumeration, mesh compression and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best known uniform random sampler for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also provides an encoding for the set $\mathcal{P}(n)$ of $n$-edge 3-connected planar graphs that matches the entropy bound $\frac1n\log_2|\m…
We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d en…
This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of α-orientations introduced by Felsner, and in particular on the existence of …
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a…
We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete three-point function converges to a simple universal scaling function, which is the continuous three-point function of pure 2D quantum gravity. We give explicit expressions for this universal three-point function both in the grand-canonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the marked vertices are a…
Thomassen in 1994 published a famous proof of the fact that the choosability of a planar graph is at most 5. Zhu in 2019 generalized this result by showing that the same bound holds for Alon-Tarsi numbers of planar graphs. We present an alternative proof of that fact, derived from the results on decompositions of planar graphs into trees known as Schnyder woods. It turns out that Thomassen's technique and our proof based on Schnyder woods have a lot in common. We discuss and explain the prominent role that counterclockwise 3-orientations play in proofs based on both these approaches.
We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained…
We compute the distance-dependent three-point function of general planar maps and of bipartite planar maps, i.e., the generating function of these maps with three marked vertices at prescribed pairwise distances. Explicit expressions are given for maps counted by their number of edges only, or by both their numbers of edges and faces. A few limiting cases and applications are discussed.
We consider the Functional Orientation 2-Color problem, which was introduced by Valiant in his seminal paper on holographic algorithms [SIAM J. Comput., 37(5), 2008]. For this decision problem, Valiant gave a polynomial time holographic algorithm for planar graphs of maximum degree 3, and showed that the problem is NP-complete for planar graphs of maximum degree 10. A recent result on defective graph coloring by Corrêa et al. [Australas. J. Combin., 43, 2009] implies that the problem is already hard for planar graphs of maximum degree 8. Together, these results leave open the hardness question…
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 a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on Delta-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), alpha-orientations of planar graphs (Felsner, resp. de …
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.…
We consider the problem of computing the distance-dependent two-point function of general planar maps and hypermaps, i.e. the problem of counting such maps with two marked points at a prescribed distance. The maps considered here may have faces of arbitrarily large degree, which requires new bijections to be tackled. We obtain exact expressions for the following cases: general and bipartite maps counted by their number of edges, 3-hypermaps and 3-constellations counted by their number of dark faces, and finally general and bipartite maps counted by both their number of edges and their number o…
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In part…
