Marc Noy
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View article: Probability of a Condorcet Winner for Large Electorates: An Analytic Combinatorics Approach
Probability of a Condorcet Winner for Large Electorates: An Analytic Combinatorics Approach Open
We study the probability that a given candidate is an alpha-winner, i.e. a candidate preferred to each other candidate j by a fraction alpha_j of the voters. This extends the classical notion of Condorcet winner, which corresponds to the c…
View article: The Erdős-Rényi Random Graph Conditioned on Every Component Being a Clique
The Erdős-Rényi Random Graph Conditioned on Every Component Being a Clique Open
Motivated by an application in community detection, we consider an \ER random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Henc…
View article: A logical limit law for $231$-avoiding permutations
A logical limit law for $231$-avoiding permutations Open
We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permuta…
View article: Enumeration of rooted 3-connected bipartite planar maps
Enumeration of rooted 3-connected bipartite planar maps Open
We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi…
View article: Cut Vertices in Random Planar Maps
Cut Vertices in Random Planar Maps Open
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 $X_n/n \to c$ in probability (for some explicit $c>0$). For so-called subcritica…
View article: Random Cubic Planar Maps
Random Cubic Planar Maps Open
We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest.From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic plana…
View article: Chordal graphs with bounded tree-width
Chordal graphs with bounded tree-width Open
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 $c,\gamma >0$ …
View article: Chordal graphs with bounded tree-width
Chordal graphs with bounded tree-width Open
Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} γ^n n!$, as $n\to\infty$, for some constants $c,γ>0$ dependin…
View article: A logical limit law for $231$-avoiding permutations
A logical limit law for $231$-avoiding permutations Open
We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $Ψ$, in the language of two total orders, the probability $p_{n,Ψ}$ that a uniform random 231-avoiding permutation o…
View article: Random cubic planar maps
Random cubic planar maps Open
We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest. From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic plan…
View article: Enumeration of rooted 3-connected bipartite planar maps
Enumeration of rooted 3-connected bipartite planar maps Open
We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi…
View article: Enumeration of rooted 3-connected bipartite planar maps
Enumeration of rooted 3-connected bipartite planar maps Open
We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi…
View article: Enumeration of chordal planar graphs and maps
Enumeration of chordal planar graphs and maps Open
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 ma…
View article: Limiting probabilities of first order properties of random sparse graphs and hypergraphs
Limiting probabilities of first order properties of random sparse graphs and hypergraphs Open
Let be the binomial random graph in the sparse regime, which as is well‐known undergoes a phase transition at . Lynch ( , 1992) showed that for every first order sentence , the limiting probability that satisfies as exists, and moreover it…
View article: Limiting probabilities of first order properties of random sparse graphs and hypergraphs
Limiting probabilities of first order properties of random sparse graphs and hypergraphs Open
"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…
View article: Cut Vertices in Random Planar Maps
Cut Vertices in Random Planar Maps Open
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 $X_n/n \to c$ in probability (for some explicit $c>0$). For so-called subcritica…
View article: Further results on random cubic planar graphs
Further results on random cubic planar graphs Open
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 coworke…
View article: On the expected number of perfect matchings in cubic planar graphs
On the expected number of perfect matchings in cubic planar graphs Open
A well-known conjecture by Lovász and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovs…
View article: Universal singular exponents in catalytic variable equations
Universal singular exponents in catalytic variable equations Open
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 coeffici…
View article: Asymptotic enumeration of labelled 4-regular planar graphs
Asymptotic enumeration of labelled 4-regular planar graphs Open
Building on previous work by the present authors [Proc. London Math. Soc. 119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs. Our estimate is of the form $g_n \sim g\cdo…
View article: Enumeration of labelled 4-regular planar graphs II: asymptotics
Enumeration of labelled 4-regular planar graphs II: asymptotics Open
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 n…
View article: Cut Vertices in Random Planar Maps
Cut Vertices in Random Planar Maps Open
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 subcritial subclasse…
View article: Further results on random cubic planar graphs
Further results on random cubic planar graphs Open
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 coworke…
View article: Random Planar Maps and Graphs with Minimum Degree Two and Three
Random Planar Maps and Graphs with Minimum Degree Two and Three Open
We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to t…