The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n verticesPostprint (author's final draft

Noy Serrano, Marcos

Requile, Clement

Rué Perna, Juan José

Electronic Notes in Discrete Mathematics

The goal of our work is to analyze random cubic planar graphs according to the
uniform distribution. More precisely, let G be the class of labelled cubic planar
graphs and let gn be the number of graphs with n vertices. Then each graph in G
with n vertices has the same probability 1/gn.Postprint (published version

UPCommons. Portal del coneixement obert de la UPC

Random cubic planar graphs revisited

Random cubic planar graphs revisitedMarc NoyDepartament de Matema`tiques, Universitat Polite`cnica de CatalunyaBarcelona Graduate School of Mathematics, SpainCle´ment Requile´Institut fu¨r Mathematik, Freie Universita¨t BerlinBerlin Mathematical School, GermanyJuanjo Rue´Departament de Matema`tiques, Universitat Polite`cnica de CatalunyaBarcelona Graduate School of Mathematics, SpainAbstractThe goal of our work is to analyze random cubic planar graphs according to theuniform distribution. More precisely, let G be the class of labelled cubic planargraphs and let gn be the number of graphs with n vertices. Then each graph in Gwith n vertices has the same probability 1/gn. This model was analyzed first byBodirsky et al. [1], and here we revisit and extend their work. The motivation forthis revision is twofold. First, some proofs in [1] where incomplete with respect tothe singularity analysis and we aim at providing full proofs. Secondly, we obtainnew results that considerably strengthen those in [1] and shed more light on thestructure of random cubic planar graphs. We present a selection of our results onasymptotic enumeration and on limit laws for parameters of random graphs.Keywords: random planar graphs, analytic combinatorics, asymptoticenumeration, limit laws1 Results on enumerationTheorem 1.1 The number cn of connected cubic planar graphs with n verticesis asymptoticallycn ∼ c · n−7/2γnn!,with c ≈ 0.030487 and γ = ρ−1 ≈ 3.132591, where ρ ≈ 0.319523 is the smallestpositive root of the equation729x12+17496x10+148716x8+513216x6−7293760x4+279936x2+46656 = 0,Theorem 1.2 The number gn of cubic planar graphs with n vertices is asymp-toticallygn ∼ g · n−7/2γnn!,where γ is as in Theorem 1.1 and g ≈ 0.030505. As a consequence, the limitingprobability p that a random cubic planar graph is connected is equal top =cg≈ 0.999397.The previous theorems were stated in [1, Theorem 2] in a less precise way andwith incomplete proofs regarding the singularity analysis. Our first goal is toprovide a full proof of these estimates. We remark that the actual value of pwas not computed in [1]. As we will see later, p can be computed exactly usingthe so-called dissymmetry theorem for tree-like structures. We also remarkthat some of the constants given in [1] are slightly incorrect.Theorem 1.3 The number hn of cubic planar multigraphs is asymptoticallyhn ∼ h · n−7/2γnmn!,with h ≈ 0.115965 and γm = ρ−1m ≈ 3.985537, where ρm ≈ 0.250907 is thesmallest positive root of729x12−17496x10+148716x8−513216x6−7293760x4−279936x2+46656 = 0.The same estimate holds for the number of connected cubic planar multigraphs,but with h replaced by the constant h′ ≈ 0.104705. The limiting probability ofconnectivity ispm =h′h≈ 0.902905.This result is also claimed in [3] without a detailed proof, but the equationdefining ρm is incorrect, as well as the claimed value γm ≈ 3.973.We remark that the proof needs again an application of the dissymmetrytheorem, since the presence of loops and multiple edges does not allow us, asfor simple graphs, to relate directly the number of graphs rooted at a vertexwith those rooted at an edge.2 Results on limit lawsOur main results in this section deal with the number of copies of a fixedsubgraph. We start with the number of triangles.Theorem 2.1 Let Xn be the number of triangles in a random cubic planargraph. Then Xn is asymptotically normal with momentsEXn ∼ µn, VarXn ∼ λn,whereµ ≈ 0.121974, λ ≈ 0.064985.It was proved in [1] that Xn is linear with high probability. Our result is aconsiderable sharpening of this fact. We wish to remark that this is the firsttime one is able to determine precisely the number of copies of a fixed graphH containing a cycle in classes of random planar graphs. The proof, basedon the so-called quasi-powers theorem, is technically involved and we are notable to extend it, for instance, to the number of quadrilaterals.Using the combinatorial decompositions introduced for counting triangleswe can enumerate triangle-free graphs.Theorem 2.2 The number tn of triangle-free cubic planar graphs with n ver-tices is asymptoticallytn ∼ t · n−7/2γnt n!,where γt ≈ 2.646686 and t > 0.Our next results concern the number of copies of graphs which are close tobeing cubic. We define a cherry as a planar graph in which all vertices havedegree 3 except for one distinguished vertex of degree 1. The smallest cherryhas 6 vertices and is obtained by subdividing an edge of K4 and adding onevertex of degree one.Theorem 2.3 Let XH,n be the number of copies of a fixed cherry H in a ran-dom cubic planar graph. Then XH,n is asymptotically normal with momentsEXH,n ∼ µn, VarXH,n ∼ ηn,where µ and η depend on the constant γ in Theorem 1.1 and H.The previous result improves on [1], where it is shown that the number ofcopies of a fixed cherry is linear w.h.p.We comment briefly on the notable differences of this model with theclassical model of random cubic (not necessarily planar) graphs [7]. A randomcubic graph is w.h.p. 3-connected. Also, the number of triangles followsa Poisson law with expectation 4/3. Finally a random cubic planar graphis 3-colorable, whereas a random cubic graph has a (small) probability ofcontaining K4 as a component, hence of being 4-chromatic.3 Discussion of the proofsAll graphs are labelled and the corresponding generating functions are expo-nential, expect for the family of rooted triangulations.The starting point is, as in [1], the enumeration of cubic 3-connected pla-nar graphs, which by duality are in bijection with planar triangulations. Tutteshowed in one of his seminal papers on the enumeration of maps [6] the (or-dinary) generating function of rooted 3-connected triangulations is given byT (z) = U(z) (1− 2U(z)) , where z = U(1− U)3.The unique singularity of T is τ = 27/256, and we have U(τ) = 1/4, T (τ) =1/8. If now M(x, y) is the GF for labelled 3-connected cubic planar graphsrooted at a directed edge, where x marks vertices and y marks edges, byduality we haveM(x, y) =12(T (x2y3)− x2y3) .(1)We define networks as in [1], but deviate slightly from the notation there.A network is a connected cubic multigraph G with an ordered pair of adjacentvertices (s, t), such that the graph obtained by removing the edge st is simple.We show that the generating function D(x) satisfies(1 +D)√(2 + x2)2 − 4x2D − 2− T (x2(1 +D)3) = 0.(2)Finally, we let C(x) be the generating function of connected cubic planargraphs. By double counting the edges, one shows that3xC ′(x) =L(x)x2+D(x)− L(x)− x2D(x)− L(x)2,(3)where L(x) is the generating function of loop networks, given byL(x) =12(2 + x2 −√(2 + x2)2 − 4x2D(x)).When solving (2) there are two possible sources of singularities for D:either a singularity coming from T or a branch point. We show that there isno branch point (this is a key point not addressed in [1]), hence the singularityρ comes from T . By eliminating from the previous equations, we show that ρis as claimed in Theorem 1.1.Next we compute the singular expansion of D(x) at ρ, which is of the formD(x) = D0 +D2X2 +D3X3 +O(X4), X =√1− x/ρ.The coefficients Di are computed from (2) and the known expansionT (z) =18− 316Z2 +√624Z3 +O(Z4), Z =√1− z/τ .Finally, plugging the singular expansion ofD into Equation (3) and integratingwe obtain an expansion for C(x). By singularity analysis (see [4]), the estimatein Theorem 1.1 follows.In order to prove Theorem 1.2, we proceed as follows. The limiting prob-ability of a random graph being connected is (see [5]) p = e−C(ρ). Hence,with the notations of theorems 1.1 and 1.2, we have cn ∼ p · gn. In order tocompute C(ρ) we need an explicit expression for C(x). The natural approachis to integrate (3), but this needs integrating D(x), and we are not able todo it algebraically. Instead we compute C(x) directly by using the dissymme-try theorem for tree-decomposable structures, as in [2]. In other words, weintegrate combinatorially, instead of algebraically. We obtain an explicit (toolong to reproduce here) expression for C(x) in terms of D(x), from which wecan compute C(ρ).The proof of the Gaussian limit law is based on the quasi-powers theorem[4]. In the case of Theorem 2.1, we add a variable u for marking triangles andwe show that the associated bivariate generating function D(x, u) of networkshas a unique dominant singularity ρ(u) and locally an expansion of the formD(x, u) = D0(u) +D2(u)X2 +D3(u)X3 +O(X4), X =√1− x/ρ(u).The probability generating function pn(u) of the random variable countingtriangles ispn(u) =[xn]D(x, u)[xn]D(x, 1).By singularity analysis we have an estimate near u = 1 (uniform on u) of theformpn(u) = A(u)B(u)n(1 +O(n−1)).Thus pn(u) is ‘close’ to being an exact power and we deduce a central limittheorem for the number of triangles. From the values pn(1) and p′n(1) onegets an estimate for the first and second moments. The main difficulty hereis to keep track of the triangles in the combinatorial decompositions and tolocate the singularities. In fact, the equations definingD(x, u) are considerablyinvolved. Once this is done, it translates easily into connected and arbitrarycubic planar graphs.Theorem 2.2 is obtained setting u = 0 in D(x, u). Since 0 is away from1, care must be taken to guarantee that, once more, there is no branch pointand the singularity comes from that of triangulations.References[1] M. Bodirsky, M. Kang, M. Lo¨ﬄer, C. McDiarmid. Random cubic planar graphs.Random Structures & Algorithms 30 (2007), 78–94.[2] G. Chapuy, E´. Fusy, M. Kang, B. Shoilekova. A complete grammar fordecomposing a family of graphs into 3-connected components. Electron. J.Combin. 15 (2008), no. 1, Research Paper 148, 39 pp.[3] W. Fang, M. Kang, M. Moßhammer, P. Spru¨ssel. Cubic graphs and relatedtriangulations on orientable surfaces (2016) arXiv:1603.01440.[4] P. Flajolet, R. Sedgewick. Analytic Combinatorics. Cambridge University Press,Cambridge, 2009.[5] O. Gime´nez, M. Noy. Asymptotic enumeration and limit laws of planar graphs.J. Amer. Math. Soc. 22 (2009), 309–329.[6] W. T. Tutte. A census of planar triangulations. Canad. J. Math. 14 (1962)21–38.[7] N. C. Wormald, Models of random regular graphs, in Surveys in Combinatorics,1999, pp. 239–298, Cambridge University Press, Cambridge (1999).

The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n vertice

UPCommons

The goal of our work is to analyze random cubic planar graphs according to the

uniform distribution. More precisely, let G be the class of labelled cubic planar

graphs and let gn be the number of graphs with n vertices. Then each graph in G

with n vertices has the same probability 1/gn

https://upcommons.upc.edu/bitstream/2117/104259/1/endm.pdf

Random cubic planar graphs revisited

Abstract

Similar works

Full text

Available Versions

UPCommons. Portal del coneixement obert de la UPC

UPCommons. Portal del coneixement obert de la UPC

UPCommons

UPCommons