Related papers: On the probability of nonexistence in binomial sub…
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.…
A non-uniform and inhomogeneous random hypergraph model is considered, which is a straightforward extension of the celebrated binomial random graph model $\mathbb G(n, p)$. We establish necessary and sufficient conditions for small…
Random subsampling of edges is a commonly employed technique in graph algorithms, underlying a vast array of modern algorithmic breakthroughs. Unfortunately, using this technique often leads to randomized algorithms with no clear path to…
Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with…
We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha+o(1)}$.
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…
Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of…
In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…
Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $\mathcal{H}$ satisfies some regularity…
Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on $n$ vertices and $m$ edges. In the first (edge-independent) model, a random hypergraph $H_1$ is constructed by fixing a…
We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.
In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree…
We find new upper bounds on the size of a minimum totally dominating set for random regular graphs and for regular graphs with large girth. These bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and…
Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…
Let $G$ be a graph on $n$ vertices and let $k$ be a fixed positive integer. We denote by $\mathcal G_{\text{$k$-out}}(G)$ the probability space consisting of subgraphs of $G$ where each vertex $v\in V(G)$ randomly picks $k$ neighbors from…
Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of $G(n,p)$ with high…
Let P_{n,m} denote the graph taken uniformly at random from the set of all planar graphs on {1,2,..., n} with exactly m(n) edges. We use counting arguments to investigate the probability that P_{n,m} will contain given components and…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from…