We study thresholds for Ramsey properties of random discrete structures. In particular, we determine the threshold for Rado's theorem for solutions of partition regular systems of equations in rand...

In this article, we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on nlabeled vertices. At each round we are presented with K = ...

We establish central and local limit theorems for the number of vertices in the largest component of a random d-uniform hypergraph Hd(n,p) with edge probability p = c-$\left(\matrix{n-1 \cr d-1 }\r...

The behavior of the random graph G(n,p) around the critical probability pc = $ {1 \over n} $ is well understood. When p = (1 + O(n1-3))pc the components are roughly of size n2-3 and converge, when ...

We study the random graph Gn,λ/n conditioned on the event that all vertex degrees lie in some given subset of the nonnegative integers. Subject to a certain hypothesis on , the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter given as the root of a certain “characteristic equation” of that maximizes a certain function . Subject to a hypothesis on , we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set including the sets of (respectively) even and odd numbers. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree.

We consider Metropolis Glauber dynamics for sampling proper q-colorings of the n-vertex complete b-ary tree when 3 ≤ q ≤ b/(2lnb). We give both upper and lower bounds on the mixing time. Our upper bound is nO(b/ log b) and our lower bound is nΩ(b/(q log b)), where the constants implicit in the O() and Ω() notation do not depend upon n, q or b. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

We consider in this article the problem of discovering, via a traceroute algorithm, the topology of a network, whose graph is spanned by an infinite branching process. A subset of nodes is selected...

We consider a fault tolerant broadcast network of n processors each holding one bit of information. The goal is to compute a given Boolean function on the n bits. In each step, a processor may broadcast one bit of information. Each listening processor receives the bit that was broadcast with error probability bounded by a fixed constant ϵ. The errors in different steps, as well as for different receiving processors in the same step, are mutually independent. The protocols that are considered in this model are oblivious protocols: At each step, the processors that broadcast are fixed in advanced and independent of the input and the outcome of previous steps. We present here the first linear complexity protocols for several classes of Boolean functions. This answer an open question of Yao (Invited talk in the 5th ISTCS Conf., 1997), considering this fault tolerant model that was introduced by El Gamal (Open problems presented at the 1984 workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research) and studied also by Gallager 10. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009