Related papers: Lack of Hyperbolicity in Asymptotic Erd\"os--Renyi…

We provide simple proofs describing the behavior of the largest component of the Erdos-Renyi random graph G(n,p) outside of the scaling window, p={1+\eps(n) \over n} where \eps(n) tends to 0, but \eps(n)n^{1/3} tends to \infty.

We provide a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. Our…

We study the near-critical behavior of the sparse Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$ on $n\gg1$ vertices, where the connection probability $p$ satisfies $np = 1+\theta(b_n^2/n)^{1/3}$, with $n^{3/10}\ll {b_n}\ll n^{1/2}$, and…

Using recent results on the concentration of the largest eigenvalue and maximal vertex degree of large random graphs, we show that the infinite sequence of Erd\H os-R\'enyi random graphs $G(n,\rho_n/n)$ such that $\rho_n/\log n$ infinitely…

In this paper we prove that random $d$--regular graphs with $d\geq 3$ have traffic congestion of the order $O(n\log_{d-1}^{3}(n))$ where $n$ is the number of nodes and geodesic routing is used. We also show that these graphs are not…

We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In…

For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t =…

For any small constant $\epsilon>0$, the Erd\H{o}s-R\'enyi random graph $G(n,\frac{1+\epsilon}{n})$ with high probability has a unique largest component which contains $(1\pm O(\epsilon))2\epsilon n$ vertices. Let $G_c(n,p)$ be obtained by…

Let $G=(V,E)$ be a connected graph with the usual (graph) distance metric $d:V \times V \to N \cup \{0 \}$. Introduced by Gromov, $G$ is $\delta$-hyperbolic if for every four vertices $u,v,x,y \in V$, the two largest values of the three…

Given a positive integer $n$, an unlabeled graph $G$ on $n$ vertices, and a vertex $v$ of $G$, let $N_G(v)$ be the subgraph of $G$ induced by vertices of $G$ of distance at most one from $v$. We show that there are universal constants…

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the…

The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…

We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely…

The largest components of the critical Erd\H{o}s-R\'enyi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of…

Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph $G(n,p)$ above the phase…

The random coloured graph $G_c(n,p)$ is obtained from the Erd\H{o}s-R\'{e}nyi binomial random graph $G(n,p)$ by assigning to each edge a colour from a set of $c$ colours independently and uniformly at random. It is not hard to see that,…

We consider the near-critical Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and $A$ is large,…

Two models of a random digraph on $n$ vertices, $D(n,\text{Prob}(\text{arc})=p)$ and $D(n,\text{number of arcs}=m)$ are studied. In 1990, Karp for $D(n,p)$ and independently T. \L uczak for $D(n,m=cn)$ proved that for $c>1$, with…

We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the…