We consider the problem of generating a coloring of the random graph 𝔾_n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ colors available, where Δ is the maximum degree of the graph, and we ...

One of the main questions that arise when studying random and quasi-random structures is which properties $\cal P$ are such that any object that satisfies $\cal P$ “behaves” like a truly random one. In the context of graphs, Chung, Graham, and Wilson (...

Derényi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph Ggenerated by some rule, form an auxiliary graph G′ whose vertices are the k-cliques of G, in ...

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 according to some criterion. As a measure of efficiency ...

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the ...

In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to ...

The even discrete torus is the graph T_L,d on vertex set {0,…,L 1}^d (with L even) in which two vertices are adjacent if they differ on exactly one coordinate and differ by 1(modL) on that coordinate. The hard-core measure with activity λ on T_L,d is the ...

We study the simple random walk on the n-dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly rescaled hitting time is asymptotically exponentially ...

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix ...

We consider the following variant of the classical random graph process introduced by Erdýs and Rényi. Starting with an empty graph on n vertices, choose the next edge uniformly at random among all edges not yet considered, but only insert it if the ...

In the on-line nearest-neighbor graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large-sample asymptotic behavior of the ...

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game ...

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two ...

In this article we study the one-dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties ...

In this paper we consider the problem of finding large collections of vertices and edges satisfying particular separation properties in random regular graphs of degree r, for each fixed r ≥ 3. We prove both constructive lower bounds and combinatorial ...

We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerdi's regularity lemma. More precisely, we consider the number of complete graphs K_ℓ on ℓ vertices in ℓ-partite graphs where each ...

We derive an expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds asymptotically almost surely, assuming that certain convergence and supercriticality conditions are met, ...

We model noisy clustering data using random graphs: Clusters correspond to disjoint sets of vertices. Two vertices from the same set (resp., different sets) share an edge with probability p (resp., r < p). We give algorithms that reconstruct the ...

Fix d≥ 2, and let X be either ℤ^d or the points of a Poisson process in ℝ^d of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a spread-out percolation model ...

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ε {1,…,n}, no label occurs twice, and whenever we proceed from the ...