Related papers: Thresholds and expectation thresholds

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges…

There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the…

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a…

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the…

For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph…

The covariance graph (aka bi-directed graph) of a probability distribution $p$ is the undirected graph $G$ where two nodes are adjacent iff their corresponding random variables are marginally dependent in $p$. In this paper, we present a…

We consider a number $\nu_n$ of components in a random graph $G(n,p)$ with $n$ vertices, where the probability of an edge is equal to $p$. By operating with special generating functions we shows the next asymptotic relation for factorial…

We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of K\"uhn and Osthus and also Nenadov and \v{S}kori\'c.

For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed…

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…

The biclique partition number of a graph $G= (V,E)$, denoted $bp(G)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that $…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle,…

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…

The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation…

In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our…

Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $\delta(H_n)\geq\alpha n$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will…

A famous conjecture of Sidorenko and Erd\H{o}s-Simonovits states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order…