Counting outerplanar maps

A map is outerplanar if all its vertices lie in the outer face. We enumerate various classes of rooted outerplanar maps with respect to the number of edges and vertices. The proofs involve several bijections with lattice paths. As a consequence of our results, we obtain an e cient scheme for encoding simple outerplanar maps.Peer ReviewedPostprint (published version

Cut vertices in random planar maps

The main goal of this paper is to determine the asymptotic behavior of the number X n of cut-vertices in random planar maps with n edges. It is shown that X n / n ¿ c in probability (for some explicit c > 0 ). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.Peer ReviewedPostprint (published version

A solution to the tennis ball problem

We present a complete solution to the so-called tennis ball problem, which is equivalent to counting the number of lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions. Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.Postprint (published version

Enumeration of chordal planar graphs and maps

We determine the number of labelled chordal planar graphs with n vertices, which is asymptotically for a constant and . We also determine the number of rooted simple chordal planar maps with n edges, which is asymptotically , where , , and s 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 rich. The 3-connected members are precisely chordal triangulations, those obtained starting from by repeatedly adding vertices adjacent to an existing triangular face.We gratefully acknowledge earlier discussions on this project with Erkan Narmanli. M.N. was supported by grants MTM2017-82166-P and PID2020-113082GB-I00, the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). C.R. was supported by the grant Beatriu de Pinós BP2019, funded by the H2020 COFUND project No 801370 and AGAUR (the Catalan agency for management of university and research grants), and the grant PID2020-113082GB-I00 of the Spanish Ministry of Science and Innovation.Peer ReviewedPostprint (author's final draft

Random cubic planar graphs revisited

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

Enumeration of labeled 4-regular planar graphs

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this extended abstract, we present the first combinatorial scheme for counting labeled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function counting labeled 4-regular planar graphs can be computed effectively as the solution of a system of equations. From here we can extract the coefficients by means of algebraic calculus. As a by-product, we can also compute the algebraic generating function counting labeled 3-connected 4-regular planar maps.Peer ReviewedPostprint (author's final draft

Enumeration of labelled 4-regular planar graphs II: asymptotics

This work is a follow-up of the article (Noy et al., 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 of labelled 4-regular planar graphs on vertices. Our estimate is of the form , where is a constant and is the radius of convergence of the generating function , and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.Peer ReviewedPostprint (author's final draft

Limiting probabilities of first order properties of random sparse graphs and hypergraphs

"This is the peer reviewed version of the following article: Larrauri, L.; Müller, T.; Noy, M. Limiting probabilities of first order properties of random sparse graphs and hypergraphs. "Random structures and algorithms", 18 Agost 2021, vol. 60, núm. 3, p. 506-526., which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/rsa.21041. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited."Let Gn be the binomial random graph G(n, p = c/n) in the sparse regime, which as is well-known undergoes a phase transition at c = 1. Lynch (Random Structures Algorithms, 1992) showed that for every first order sentence f, the limiting probability that Gn satisfies f as n ¿ 8 exists, and moreover it is an analytic function of c. In this paper we consider the closure Lc in [0, 1] of the set Lc of all limiting probabilities of first order sentences in Gn. We show that there exists a critical value c0 ˜ 0.93 such that Lc = [0, 1] when c = c0, whereas Lc misses at least one subinterval when c < c0. We extend these results to random d-uniform sparse hypergraphs, where the probability of a hyperedge is given by p = c/nd-1 .Peer ReviewedPostprint (author's final draft

Further results on random cubic planar graphs

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. 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 asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.Peer ReviewedPostprint (author's final draft

Chordal graphs with bounded tree-width

Given $t\ge 2$ and $0\le k\le t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} \gamma^n n!$, as $n\to\infty$, for some constants $c,\gamma >0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\le i\le t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$. The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\ge 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, \textit{Graphs and Combinatorics} (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices.Peer ReviewedPostprint (author's final draft