On the expected number of perfect matchings in cubic planar graphs

  title={On the expected number of perfect matchings in cubic planar graphs},
  author={Marc Noy and Cl'ement Requil'e and Juanjo Ru'e},
A well-known conjecture by Lovasz 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, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the… 

The scaling limit of random cubic planar graphs

— We study the random simple connected cubic planar graph Cn with an even number n of vertices. We show that the Brownian map arises as Gromov– Hausdorff–Prokhorov scaling limit of Cn as n ∈ 2N tends

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The main result shows that the first-passage percolation distance concentrates in an op(n ) window around a constant multiple of the graph distance.

What is a random surface?

Given 2n unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a

The Uniform Infinite Cubic Planar Graph

— We prove that the random simple connected cubic planar graph Cn with an even number n of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how



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It is proved that if G is a planar cubic graph with no cutedge, then G has at least 2-V (G), which is the number of perfect matchings in G.

Further results on random cubic planar graphs

To the best of the 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.

Asymptotic enumeration of labelled 4-regular planar graphs

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

Random cubic planar graphs

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The Numbers of Spanning Trees, Hamilton Cycles and Perfect Matchings in a Random Graph

  • S. Janson
  • Mathematics
    Combinatorics, Probability and Computing
  • 1994
The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph G nm are shown to be asymptotically normal if m is neither too large nor too small, and the results are proved using decomposition and projection methods.

Random planar graphs and beyond

We survey several results on the enumeration of planar graphs and on properties of random planar graphs. This includes basic parameters, such as the number of edges and the number of connected

Asymptotic enumeration and limit laws of planar graphs

A graph is planar if it can be embedded in the plane, or in the sphere, so that no two edges cross at an interior point. A planar graph together with a particular embedding is called a map. There is