What is the probability that the number of triangles in , the Erdős-Rényi random graph with edge density *p*, is at least twice its mean? Writing it as , already the order of the rate function *r*(*n, p*) was a longstanding open problem when *p* = *o*(1), finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that for ; the exact asymptotics of *r*(*n, p*) remained unknown. The following variational problem can be related to this large deviation question at : for *δ* > 0 fixed, what is the minimum asymptotic *p*-relative entropy of a weighted graph on *n* vertices with triangle density at least (1 + *δ*)*p*^{3}? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this problem for *fixed p*. A very recent breakthrough of Chatterjee and Dembo extended its validity to for an explicit *α* > 0, and plausibly it holds in all of the above sparse regime.

In this note we show that the solution to the variational problem is when vs. when (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that for has twice as many triangles as its expectation is where . Our results further extend to *k*-cliques for any fixed *k*, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

A graph *G* is said to be -universal if it contains every graph on at most *n* vertices with maximum degree at most Δ. It is known that for any and any natural number Δ there exists such that the random graph *G*(*n, p*) is asymptotically almost surely -universal for . Bypassing this natural boundary, we show that for the same conclusion holds when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Consider a homogeneous Poisson point process in a compact convex set in *d*-dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Basic graph structures such as maximal independent sets (MIS's) have spurred much theoretical research in randomized and distributed algorithms, and have several applications in networking and distributed computing as well. However, the extant (distributed) algorithms for these problems do not necessarily guarantee fault-tolerance or load-balance properties. We propose and study “low-average degree” or “sparse” versions of such structures. Interestingly, in sharp contrast to, say, MIS's, it can be shown that checking whether a structure is sparse, will take substantial time. Nevertheless, we are able to develop good sequential/distributed (randomized) algorithms for such sparse versions. We also complement our algorithms with several lower bounds. Randomization plays a key role in our upper and lower bound results. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let *G* be a connected bounded-degree graph. Given an edge *e* in *G* we would like to decide whether *e* belongs to a connected subgraph consisting of edges (for a prespecified constant ), where the decision for different edges should be consistent with the same subgraph . Can this task be performed by inspecting only a *constant* number of edges in *G*? Our main results are:

- We show that if every
*t*-vertex subgraph of*G*has expansion then one can (deterministically) construct a sparse spanning subgraph of*G*using few inspections. To this end we analyze a “local” version of a famous minimum-weight spanning tree algorithm. - We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every
*t*-vertex subgraph has expansion . We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.

© 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Suppose that are independent identically distributed Bernoulli random variables with mean *p*, so and . Any estimate of *p* has relative error . This paper builds a new estimate of *p* with the remarkable property that the relative error of the estimate does not depend in any way on the value of *p*. This allows the easy construction of exact confidence intervals for *p* of any desired level without needing any sort of limit or approximation. In addition, is unbiased. For *∊* and *δ* in (0, 1), to obtain an estimate where , the new algorithm takes on average at most samples. It is also shown that any such algorithm that applies whenever requires at least samples on average. The same algorithm can also be applied to estimate the mean of any random variable that falls in . The used here employs randomness external to the sample, and has a small (but nonzero) chance of being above 1. It is shown that any nontrivial where the relative error is independent of *p* must also have these properties. Applications of this methodology include finding exact *p*-values and randomized approximation algorithms for # *P* complete problems. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer *k* and let *G*_{k} be the random subgraph where each independently chooses *k* random neighbors, making *kn* edges in all. When the minimum degree then *G*_{k} is *k*-connected w.h.p. for ; Hamiltonian for *k* sufficiently large. When , then *G*_{k} has a cycle of length for . By w.h.p. we mean that the probability of non-occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension *d*, a random walk with an absorbing state is defined which relates to the spectrum of the *k*-dimensional Laplacian for 1 ≤ *k* ≤ *d*. We study an example of random walks on simplicial complexes in the context of a semi-supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Let *T* be a regular rooted tree. For every natural number *n*, let *T*_{n} be the finite subtree of vertices with graph distance at most *n* from the root. Consider the following forest-fire model on *T*_{n}: Each vertex can be “vacant” or “occupied”. At time 0 all vertices are vacant. Then the process is governed by two opposing mechanisms: Vertices become occupied at rate 1, independently for all vertices. Independently thereof and independently for all vertices, “lightning” hits vertices at rate *λ*(*n*) > 0. When a vertex is hit by lightning, its occupied cluster becomes vacant instantaneously. Now suppose that *λ*(*n*) decays exponentially in *n* but much more slowly than 1/|*T*_{n}|, where |*T*_{n}| denotes the number of vertices of *T*_{n}. We show that then there exist such that between time 0 and time the forest-fire model on *T*_{n} tends to the following process on *T* as *n* goes to infinity: At time 0 all vertices are vacant. Between time 0 and time *τ* vertices become occupied at rate 1, independently for all vertices. Immediately before time *τ* there are infinitely many infinite occupied clusters. At time *τ* all these clusters become vacant. Between time *τ* and time vertices again become occupied at rate 1, independently for all vertices. At time all occupied clusters are finite. This process is a dynamic version of self-destructive percolation. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three-fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank-based selection model, the Aiello-Chung-Lu models, the generalized linear preference model, directed scale-free graphs, the Cooper-Frieze model, and random unordered increasing *k*-trees. Our results shed light on why the small-world phenomenon is observed in so many real-world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edges essentially matching results for Erdős–Rényi digraphs obtained from Chernoff-type bounds. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee in (Chatterjee, Probab Theory and Relat Fields 138 (2007), 305–321). Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well-known “simple switching” operation, as well as a novel “reflection” operation. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

In 2002, Bollobás and Scott posed the following problem: for an integer and a graph *G* of *m* edges, what is the smallest *f*(*k, m*) such that *V*(*G*) can be partitioned into *V* _{1},…,*V*_{k} in which for all , where denotes the number of edges with both ends in ? In this paper, we solve this problem asymptotically by showing that . We also show that *V*(*G*) can be partitioned into such that for , where Δ denotes the maximum degree of *G*. This confirms a conjecture of Bollobás and Scott. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We show that, for a natural notion of quasirandomness in *k*-uniform hypergraphs, any quasirandom *k*-uniform hypergraph on *n* vertices with constant edge density and minimum vertex degree Ω(*n*^{k-1}) contains a loose Hamilton cycle. We also give a construction to show that a *k*-uniform hypergraph satisfying these conditions need not contain a Hamilton *ℓ*-cycle if *k*– *ℓ* divides *k*. The remaining values of *ℓ* form an interesting open question. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant spanning ratio (w.r.t the Euclidean distance) which follows vertex adjacencies in the Delaunay triangulation. We call this strategy *cone walk*. We prove that given *n* uniform points in a smooth convex domain of unit area, and for any start point *z* and query point *q*; cone walk applied to *z* and *q* will access at most sites with complexity with probability tending to 1 as *n* goes to infinity. We additionally show that in this model, cone walk is -memoryless with high probability for any pair of start and query point in the domain, for any positive *ξ*. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet, we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs and whether asymptotically good linear LTCs with constant query complexity exist. In this paper, we provide a combinatorial characterization of smooth locally testable codes, which are locally testable codes whose associated tester queries every bit of the tested word with equal probability. Our main contribution is a combinatorial property defined on the Tanner graph associated with the code tester (“well-structured tester”). We show that a family of codes is smoothly locally testable if and only if it has a well-structured tester.

As a case study we show that the standard tester for the Hadamard code is “well-structured,” giving an alternative proof of the local testability of the Hadamard code, originally proved by (Blum, Luby, Rubinfeld, J. Comput. Syst. Sci. 47 (1993) 549–595) (STOC 1990). Additional connections to the works of (Ben-Sasson, Harsha, Raskhodnikova, SIAM J. Comput 35 (2005) 1–21) (SICOMP 2005) and of (Lachish, Newman and Shapira, Comput Complex 17 (2008) 70–93) are also discussed. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We study testing properties of functions on finite groups. First we consider functions of the form , where *G* is a finite group. We show that conjugate invariance, homomorphism, and the property of being proportional to an irreducible character is testable with a constant number of queries to *f*, where a character is a crucial notion in representation theory. Our proof relies on representation theory and harmonic analysis on finite groups. Next we consider functions of the form , where *d* is a fixed constant and is the family of *d* by *d* matrices with each element in . For a function , we show that the unitary isomorphism to *g* is testable with a constant number of queries to *f*, where we say that *f* and *g* are unitary isomorphic if there exists a unitary matrix *U* such that for any . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to *minimize the expected value of the time* that is needed to complete the selection of a monotone increasing subsequence of a prespecified length *n*. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability , the random graph *G*(*n, p*) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph , an *incompatibility system* over *G* is a family where for every , the set *F*_{v} is a set of unordered pairs . An incompatibility system is *Δ-bounded* if for every vertex *v* and an edge *e* incident to *v*, there are at most Δ pairs in *F*_{v} containing *e*. We say that a cycle *C* in *G* is *compatible* with if every pair of incident edges of *C* satisfies . This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant such that the random graph with is asymptotically almost surely such that for any *μnp*-bounded incompatibility system over *G*, there is a Hamilton cycle in *G* compatible with . We also prove that for larger edge probabilities , the parameter *μ* can be taken to be any constant smaller than . These results imply in particular that typically in *G*(*n, p*) for , for any edge-coloring in which each color appears at most *μnp* times at each vertex, there exists a properly colored Hamilton cycle. Furthermore, our proof can be easily modified to show that for any edge-coloring of such a random graph in which each color appears on at most *μnp* edges, there exists a Hamilton cycle in which all edges have distinct colors (i.e., a rainbow Hamilton cycle). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

In this paper, we prove a local limit theorem for the distribution of the number of triangles in the Erdos-Rényi random graph *G*(*n, p*), where is a fixed constant. Our proof is based on bounding the characteristic function of the number of triangles, and uses several different conditioning arguments for handling different ranges of *t*. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We show that any *k*-uniform hypergraph with *n* edges contains two isomorphic edge disjoint subgraphs of size for *k* = 4, 5 and 6. This is best possible up to a logarithmic factor due to an upper bound construction of Erdős, Pach, and Pyber who show there exist *k*-uniform hypergraphs with *n* edges and with no two edge disjoint isomorphic subgraphs with size larger than . Furthermore, our result extends results Erdős, Pach and Pyber who also established the lower bound for *k* = 2 (eg. for graphs), and of Gould and Rödl who established the result for *k* = 3. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

A random spherical polytope *P*_{n} in a spherically convex set as considered here is the spherical convex hull of *n* independent, uniformly distributed random points in *K*. The behaviour of *P*_{n} for a spherically convex set *K* contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when *K* is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as *n* tends to infinity, of the expectation of several characteristics of *P*_{n}, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is on an *n*-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is and . We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of *K*_{n} is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large *p*, every maximum triangle-free subgraph of *G*(*n, p*) is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for for some constant *K*, and apart from the value of the constant this bound is best possible.

We study an extremal problem of this type in random hypergraphs. Denote by *F*_{5}, which is sometimes called the generalized triangle, the 3-uniform hypergraph with vertex set and edge set . One of the first results in extremal hypergraph theory is by Frankl and Füredi, who proved that the maximum 3-uniform hypergraph on *n* vertices containing no copy of *F*_{5} is tripartite for *n* > 3000. A natural question is for what *p* is every maximum *F*_{5}-free subhypergraph of w.h.p. tripartite. We show this holds for for some constant *K* and does not hold for . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdős-Rényi random graph *G*(*n, p*). Under the alternative, the graph is generated from the model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere , and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., *p* is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group *S*_{n} generate a transitive subgroup with probability for some independent of *n*, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of *A*_{n} or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We study the Maker-Breaker *H*-game played on the edge set of the random graph . In this game two players, Maker and Breaker, alternately claim unclaimed edges of , until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph *H*; Breaker wins otherwise.

In this paper we show that, with the exception of trees and triangles, the threshold for an *H*-game is given by the threshold of the corresponding Ramsey property of with respect to the graph *H*. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on *n* vertices with minimum out-degree and in-degree at least contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph , that is, a directed graph in which every ordered pair (*u, v*) becomes an arc with probability *p* independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if , then a.a.s. every subdigraph of with minimum out-degree and in-degree at least contains a directed Hamilton cycle. The constant 1/2 is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures.

In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non-tree’ evolutions where cycles may develop in the network.

A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through *C*_{0}-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses vertices according to a preferential rule and connects to the vertex in the selection with the th highest degree. For meek choice, where , we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where , we confirm that the degree distribution asymptotically follows a power law with logarithmic correction when and shows condensation-like behaviour when . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We compute an asymptotic expansion in of the limit in of the empirical spectral measure of the adjacency matrix of an Erdős-Rényi random graph with vertices and parameter . We present two different methods, one of which is valid for the more general setting of locally tree-like graphs. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

The Push-Pull protocol is a well-studied round-robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random -trees, a class of power law graphs, which are small-world and have large clustering coefficients, built as follows: initially we have a -clique. In every step a new node is born, a random -clique of the current graph is chosen, and the new node is joined to all nodes of the -clique. When is fixed, we show that if initially a random node is aware of the rumor, then with probability after rounds the rumor propagates to nodes, where is the number of nodes and is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion and constant treewidth, these results demonstrate that Push-Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability the protocol needs at least rounds to inform all nodes. This exponential dichotomy between time required for informing *almost all* and *all* nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random -Apollonian networks, for which we prove an upper bound of rounds for informing nodes with probability when is fixed. Here, © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Two models of a random digraph on *n* vertices, and are studied. In 1990, Karp for *D*(*n, p*) and independently T. Łuczak for *D*(*n,m* = *cn*) proved that for *c* > 1, with probability tending to 1, there is an unique strong component of size of order *n*. Karp showed, in fact, that the giant component has likely size asymptotic to *nθ*^{2}, where *θ* = *θ*(*c*) is the unique positive root of . In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector , and two distinct 2 × 2 covariance matrices, and . To this end, we introduce and analyze a randomized deletion process which terminates at the directed (1, 1)-core, the maximal digraph with minimum in-degree and out-degree at least 1. This (1, 1)-core contains all non-trivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the (1, 1)-core, those outside the largest strong component, are of polylog order, thus dwarfed by anticipated fluctuations, on the scale of *n*^{1/2}, of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier-based techniques. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We show that if , then is -close to a junta depending upon at most coordinates, where denotes the edge-boundary of in the -grid. This bound is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [6], or as a characterisation of large subsets of the -grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [1]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

For let denote the tree consisting of an -vertex path with disjoint -vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of . We derive exact formulas for the probability that the *i*^{th} element of a random permutation is a specific value less than *i*, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large *N* when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312-avoiding permutation has *k* specified decreasing points, and we show that for large *N* the points below the diagonal look like trajectories of a random walk. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We study Bernoulli bond percolation on a random recursive tree of size *n* with percolation parameter *p*(*n*) converging to 1 as *n* tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree-indexed process of cluster sizes to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous-time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

*Abstract–*We study a natural process for allocating balls into bins that are organized as the vertices of an undirected graph . Balls arrive one at a time. When a ball arrives, it first chooses a vertex in uniformly at random. Then the ball performs a local search in starting from until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case , we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the *cover time* of this process, which is defined as the smallest so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

We present an approximation algorithm for -instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio . In other words, given a -edge-weighting of the complete graph on vertices, our algorithm outputs a Hamilton cycle of with the following property: the proportion of Hamilton cycles of whose weight is smaller than that of is at most . Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant such that cannot be replaced by in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 454–479, 2016

We study the random variable *B*(*c, n*), which counts the number of balls that must be thrown into *n* equally-sized bins in order to obtain *c* collisions. The asymptotic expected value of *B*(1, *n*) is the well-known appearing in the solution to the birthday problem; the limit distribution and asymptotic moments of *B*(1, *n*) are also well known. We calculate the distribution and moments of *B*(*c, n*) asymptotically as *n* goes to *∞* and *c* = *O*(*n*). We have two main tools: an embedding of the collision process — realizing the process as a deterministic function of the standard Poisson process — and a central limit result by Rényi. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 480–502, 2016

Let be drawn uniformly from all *m*-edge, *k*-uniform, *k*-partite hypergraphs where each part of the partition is a disjoint copy of . We let be an edge colored version, where we color each edge randomly from one of colors. We show that if and where *K* is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if *n* is even and where *K* is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in . Here denotes a random edge coloring of with *n* colors. When *n* is odd, our proof requires for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016

Given an undirected *n*-vertex graph *G*(*V, E*) and an integer *k*, let *T*_{k}(*G*) denote the random vertex induced subgraph of *G* generated by ordering *V* according to a random permutation *π* and including in *T*_{k}(*G*) those vertices with at most *k* – 1 of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the *layers model with parameter k*. The layers model has found applications in studying *ℓ*-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are 3-regular graphs *G* for which with high probability *T*_{3}(*G*) has a connected component of size , and moreover, *T*_{3}(*G*) has treewidth . In contrast, *T*_{2}(*G*) is known to be a forest (hence of treewidth 1), and we prove that if *G* is of bounded degree then with high probability the largest connected component in *T*_{2}(*G*) is of size . We also consider the infinite grid , for which we prove that contains a unique infinite connected component with probability 1. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 524–545, 2016

In this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the *maximum density* of a graph *H* be the maximum average degree of all the subgraphs of *H*. First, we show that for , a graph w.h.p. contains copies of all spanning graphs *H* with maximum degree at most Δ and maximum density at most *d*. For , this improves a result of Dellamonica, Kohayakawa, Rödl and Rucińcki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij.

Finally, in the same spirit, we show that for any spanning graph *H* with constant maximum degree, and for suitable *p*, if we randomly color the edges of a graph with colors, then w.h.p. there exists a *rainbow* copy of *H* in *G* (that is, a copy of *H* with all edges colored with distinct colors). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 546–564, 2016

We study randomized gossip-based processes in dynamic networks that are motivated by information discovery in large-scale distributed networks such as peer-to-peer and social networks. A well-studied problem in peer-to-peer networks is *resource discovery*, where the goal for nodes (hosts with IP addresses) is to discover the IP addresses of all other hosts. Also, some of the recent work on self-stabilization algorithms for P2P/overlay networks proceed via discovery of the complete network. In social networks, nodes (people) discover new nodes through exchanging contacts with their neighbors (friends). In both cases the discovery of new nodes changes the underlying network — new edges are added to the network — and the process continues in the changed network. Rigorously analyzing such dynamic (stochastic) processes in a continuously changing topology remains a challenging problem with obvious applications.

This paper studies and analyzes two natural gossip-based discovery processes. In the *push discovery* or *triangulation* process, each node repeatedly chooses two random neighbors and connects them (i.e., “pushes” their mutual information to each other). In the *pull discovery* process or the *two-hop walk*, each node repeatedly requests or “pulls” a random contact from a random neighbor and connects itself to this two-hop neighbor. Both processes are lightweight in the sense that the amortized work done per node is constant per round, local, and naturally robust due to the inherent randomized nature of gossip.

Our main result is an almost-tight analysis of the time taken for these two randomized processes to converge. We show that in any undirected *n*-node graph both processes take rounds to connect every node to all other nodes with high probability, whereas is a lower bound. We also study the two-hop walk in directed graphs, and show that it takes time with high probability, and that the worst-case bound is tight for arbitrary directed graphs, whereas Ω(*n*^{2}) is a lower bound for strongly connected directed graphs. A key technical challenge that we overcome in our work is the analysis of a randomized process that itself results in a constantly changing network leading to complicated dependencies in every round. We discuss implications of our results and their analysis to discovery problems in P2P networks as well as to evolution in social networks. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 48, 565–587, 2016

Let *f* be an edge ordering of *K*_{n}: a bijection . For an edge , we call *f*(*e*) the label of *e*. An *increasing path* in *K*_{n} is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let *S*(*f*) be the number of edges in the longest increasing path. Chvátal and Komlós raised the question of estimating *m*(*n*): the minimum value of *S*(*f*) over all orderings *f* of *K*_{n}. The best known bounds on *m*(*n*) are , due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades.

In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, *S*(*f*) often takes its maximum possible value of *n* – 1 (visiting all of the vertices with an increasing Hamiltonian path). We prove that this occurs with probability at least about 1/ *e*. We also prove that with probability 1- *o*(1), there is an increasing path of length at least 0.85 *n*, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 588–611, 2016

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of for the two-sided-error randomized decision tree complexity of evaluating height *h* formulae with error . This improves the lower bound of given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most . The previous best known algorithm achieved complexity . The new lower bound follows from a better analysis of the base case of the recursion of Jayram *et al*. The new algorithm uses a novel “interleaving” of two recursive algorithms. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 612–638, 2016