Related papers: Brownian approximation to counting graphs

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with…

We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \[ \operatorname{Var}_x(\tau_y)+\mathbb E_x\tau_y \ge \frac{(\mathbb E_x\tau_y)^2}{1+\log n}, \]…

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\Bbb R}^d$. We present combinatorial expressions for the computation of the cumulants and…

We construct Brownian motion on a wide class of metric spaces similar to graphs, and show that its cover time admits an upper bound depending only on the length of the space.

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of…

We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations,…

A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio…

Given query access to an undirected graph $G$, we consider the problem of computing a $(1\pm\epsilon)$-approximation of the number of $k$-cliques in $G$. The standard query model for general graphs allows for degree queries, neighbor…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

We give the first almost-linear time algorithm for computing the \emph{maximal $k$-edge-connected subgraphs} of an undirected unweighted graph for any constant $k$. More specifically, given an $n$-vertex $m$-edge graph $G=(V,E)$ and a…

The $k$-truss, introduced by Cohen (2005), is a graph where every edge is incident to at least $k$ triangles. This is a relaxation of the clique. It has proved to be a useful tool in identifying cohesive subnetworks in a variety of…

In 2023, Gollin, Hendrey, Methuku, Tompkins and Zhang determined the maximum number of cliques in general 1-planar graphs with order $n$. Their extremal examples have connectivity at most three, except for a few small orders. At the…

Let $M_n$ be a simple triangulation of the sphere $S^2$, drawn uniformly at random from all such triangulations with n vertices. Endow $M_n$ with the uniform probability measure on its vertices. After rescaling graph distance on $V(M_n)$ by…

We show that the maximum number of triples on $n$~points, if no three triples span at most five points, is $(1\pm o(1))n^2/5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices…

We consider graph classes $\mathcal G$ in which every graph has components in a class $\mathcal{C}$ of connected graphs. We provide a framework for the asymptotic study of $\lvert\mathcal{G}_{n,N}\rvert$, the number of graphs in…

Consider a stationary Poisson point process in $\mathbb{R}^d$ and connect any two points whenever their distance is less than or equal to a prescribed distance parameter. This construction gives rise to the well known random geometric…

We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that…

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the…

The colored neighborhood metric for sparse graphs was introduced by Bollob\'as and Riordan. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm. We prove that even in this refined sense, the…

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…