Related papers: The Uniform Infinite Cubic Planar Graph

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We…

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=\omega(N^{1/3}\ln^{2} N)$. This almost completely fills the gap…

The classical definitions of the Incipient Infinite Cluster (IIC) of percolation consist in conditioning the origin on being connected to radius $n$ and letting $n$ go to infinity. We provide a short proof of that convergence in the planar…

We prove that, for every natural number $k$, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in $[k,2k+9]$. We also show that this bound is close to being optimal by constructing, for every even $k\geq…

We prove that every locally Hamiltonian graph with $n\ge 3$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a…

We prove the existence of the local limit of uniform random d-regular bipartite planar maps, for every $d\geq 3$, as the number of vertices tends to infinity. The proof relies on a bijection between maps and so-called blossoming trees…

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C*-correspondence, and from this correspondence one may…

Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every $2$-arc lies in a unique quadrangle. A graph $\Gamma$ is locally rank 3 if there exists $G\leq…

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak…

We define notions of local topological convergence and local geometric convergence for embedded graphs in $\mathbb{R}^n,$ and study their properties. The former is related to Benjamini-Schramm convergence, and the latter to weak convergence…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map (CRUM) of a given genus that start with a suitably tilted Brownian continuum…

The Blow-up Lemma established by Koml\'os, S\'ark\"ozy, and Szemer\'edi in 1997 is an important tool for the embedding of spanning subgraphs of bounded maximum degree. Here we prove several generalisations of this result concerning the…

This article aims to develop the uniformization and boundary theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) and their dual 1-skeleton, i.e., ideal angled graphs (abbr. IAG) from multiple perspectives, including…

We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of…

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the…

We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann-distributed limit structure. We demon- strate how this setting encompasses arbitrary weighted…

We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval…