Related papers: Counting connected hypergraphs via the probabilist…

Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer\'edi and Petruska proved $n\leq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n…

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n,r). Erd\H{o}s and Lov\'{a}sz conjectured that m(n,2)=\theta(n 2^n)$. The best known lower bound m(n,2)=\Omega(sqrt(n/log(n)) 2^n) was…

Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…

In 1973 P. Erd\H{o}s and L. Lov\'asz noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erd\H{o}s and Lov\'asz proved that…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with $n$ nodes is constructed as…

A $k$-uniform hypergraph is $s$-almost intersecting if every edge is disjoint from exactly $s$ other edges. Gerbner, Lemons, Palmer, Patk\'os and Sz\'ecsi conjectured that for every $k$, and $s>s_0(k)$, every $k$-uniform $s$-almost…

In this paper, we consider the hypothesis testing of correlation between two $m$-uniform hypergraphs on $n$ unlabelled nodes. Under the null hypothesis, the hypergraphs are independent, while under the alternative hypothesis, the hyperdges…

Consider a `dense' Erd\H{o}s--R\'enyi random graph model $G=G_{n,M}$ with $n$ vertices and $M$ edges, where we assume the edge density $M/\binom{n}{2}$ is bounded away from 0 and 1. Fix $k=k(n)$ with $k/n$ bounded away from 0 and~1, and let…

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…

The \emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward…

We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with…

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and…

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, ie…

We consider the infinite directed graph with vertices the set of integers ...,-2,-1,0,1,2,... . Let v be a random variable taking either finite values or value "minus infinity". Consider random weights v(j,k), indexed by pairs (j,k) of…

As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an…

The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete…

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…

One of the major problems in combinatorics is to determine the number of $r$-uniform hypergraphs ($r$-graphs) on $n$ vertices which are free of certain forbidden structures. This problem dates back to the work of Erd\H{o}s, Kleitman and…

Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple…