On the limiting distribution of the metric dimension for random forests

The metric dimension of a graph G is the minimum size of a subset S of vertices of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension for two different models of random forests, in each case obtaining normal limit distributions for this parameter.Comment: 22 pages, 5 figure

Many 2-level polytopes from matroids

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent (n-1)-dimensional 2-level polytopes is bounded from below by $c \cdot n^{-5/2} \cdot \rho^{-n}$, where $c\approx 0.03791727$ and $\rho^{-1} \approx 4.88052854$.Comment: revised version, 19 pages, 7 figure

Domino tilings of the Aztec diamond

Imagine you have a cutout from a piece of squared paper and a pile of dominoes, each of which can cover exactly two squares of the squared paper. How many different ways are there to cover the entire paper cutout with dominoes? One specific paper cutout can be mathematically described as the so-called Aztec Diamond, and a way to cover it with dominoes is a domino tiling. In this snapshot we revisit some of the seminal combinatorial ideas used to enumerate the number of domino tilings of the Aztec Diamond. The existing connection with the study of the so-called alternating-sign matrices is also explored

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz 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 expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

Threshold functions and Poisson convergence for systems of equations in random sets

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "$\mathcal{A}$ contains a non-trivial solution of $M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each of its elements is chosen independently with the same probability from the interval of integers $\{1,\dots,n\}$. Our study contains a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page

Enumeration of labelled 4-regular planar graphs

We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps

On the probability of planarity of a random graph near the critical point

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. For each $\lambda$, we find an exact analytic expression for $$p(\lambda) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+\lambda n^{-1/3})) is planar}.$$ In particular, we obtain $p(0) \approx 0.99780$. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of $G(n,\textstyle{n\over 2})$ being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur