Related papers: Edge correlations in random regular hypergraphs an…
In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…
In this paper we continue the study of a natural generalization of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that…
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results…
In this paper, we revisit the problem of sampling edges in an unknown graph $G = (V, E)$ from a distribution that is (pointwise) almost uniform over $E$. We consider the case where there is some a priori upper bound on the arboriciy of $G$.…
Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of…
In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph $K_n$ are ordered uniformly at random and are then revealed consecutively to a player…
For positive integers $r > \ell$, an $r$-uniform hypergraph is called an $\ell$-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of $r$ consecutive vertices, and such that every pair of…
We show that provided $\log^{50} n/n \leq p \leq 1 - n^{-1/4}\log^9 n$ we can with high probability find a collection of $\lfloor \delta(G)/2 \rfloor$ edge-disjoint Hamilton cycles in $G \sim G_{n, p}$, plus an additional edge-disjoint…
For positive integers $d<k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says…
Random hypergraphs extend the classical notion of random graphs by allowing hyperedges to join more than two vertices, making them well-suited for modeling higher-order interactions in complex systems. Despite their broad applicability,…
Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices…
Correlation analysis is a fundamental problem in statistics. In this paper, we consider the correlation detection problem between a pair of Erdos-Renyi graphs. Specifically, the problem is formulated as a hypothesis testing problem: under…
In a celebrated work, Blais, Brody, and Matulef developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower…
We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance…
We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs, vertex random graphs are generalizations of geometric random graphs, and…
Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world…
For a graph $H$, let $m(H,k)$ denote the number of vertices of degree $k$ in $H$. A conjecture of Alon and Wei states that for any $d\geq 3$, every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying…
Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We…
A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…
Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's…