Counting dense connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on with m edges, whenever and . We give an asymptotic formula for the number of connected r‐uniform hypergraphs on with m edges, whenever is fixed and with , that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja‐Oghlan, and Kang (the case ) and the present authors (the case , ie, “nullity” or “excess” o(n)). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use “smoothing” techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.     

σ‐algebras for quasirandom hypergraphs

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of . We associate each notion of quasirandomness with a collection of hypergraphs, the ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ‐adapted hypergraph. We then identify, for each , a particular ‐adapted hypergraph with the property that if G contains roughly the correct number of copies of then G is quasirandom in the sense of . This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017     

A note on the random greedy independent set algorithm

Let r be a fixed constant and let be an r‐uniform, D‐regular hypergraph on N vertices. Assume further that for some . Consider the random greedy algorithm for forming an independent set in . An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and contains no edge in ). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of ; that is, the process terminates at a maximal independent set. We prove that if satisfies certain degree and codegree conditions then there are vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H‐free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016     

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least infected neighbours becomes infected and remains so forever. Assume that initially vertices are randomly infected, where is the total number of vertices of the graph. Suppose also that , where is the average degree. We determine a critical function such that when , complete infection occurs with high probability as , but when , then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to .     

On the total variation distance between the binomial random graph and the random intersection graph

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .     

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

Two models of a random digraph on n vertices, and are studied. In 1990, Karp for D(n, p) and independently T. ?uczak 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 nθ², 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., 49, 3–64, 2016     

On replica symmetry of large deviations in random graphs

The following question is due to Chatterjee and Varadhan (2011). Fix and take , the Erd?s‐Rényi random graph with edge density p, conditioned to have at least as many triangles as the typical . Is G close in cut‐distance to a typical ? Via a beautiful new framework for large deviation principles in , Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d‐regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all such that lies on the convex minorant of where is the rate function of a binomial with parameter p. In particular, the answer for triangles involves rather than the natural guess of where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of is conditioned to exceed the typical value of the largest eigenvalue of . Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109–146, 2015     

Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time

Let be a connected graph with vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge‐process (or E‐process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E‐process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap of the transition matrix of a simple random walk on G. A vertex v is ?‐good, if any even degree subgraph containing all edges incident with v contains at least ? vertices. A graph G is ?‐good, if every vertex has the ?‐good property. Let G be an even degree ?‐good expander of bounded maximum degree. Any E‐process on G has vertex cover time This is to be compared with the lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on‐line by an adversary. As no walk based process can cover an n vertex graph in less than n – 1 steps, the cover time of the E‐process is of optimal order when . With high probability random r‐regular graphs, even, have . Thus the vertex cover time of the E‐process on such graphs is . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 36–54, 2015     

On MAXCUT in strictly supercritical random graphs,and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky, and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of in terms of ɛ. We then apply this result to prove the following conjecture by Frieze and Pegden. For every , there exists such that w.h.p. is not homomorphic to the cycle on vertices. We also consider the coloring properties of biased random tournaments. A p‐random tournament on n vertices is obtained from the transitive tournament by reversing each edge independently with probability p. We show that for the chromatic number of a p‐random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case we use the aforementioned result on MAXCUT and show that in fact w.h.p. one needs to reverse edges to make it 2‐colorable.     

Meyniel's conjecture holds for random graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph , which improves upon existing results showing that asymptotically almost surely the cop number of is provided that for some . We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. This will also be used in a separate paper on random d‐regular graphs, where we show that the conjecture holds asymptotically almost surely when . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396–421, 2016     

Generating random graphs in biased Maker‐Breaker games

We present a general approach connecting biased Maker‐Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker‐Breaker games. In particular, we show that for , Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a game on . As another application, we show that for , playing a game on , Maker can build a graph which contains copies of all spanning trees having maximum degree with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 47, 615–634, 2015     

Chromatic thresholds in sparse random graphs

The chromatic threshold of a graph H with respect to the random graph G (n, p ) is the infimum over d > 0 such that the following holds with high probability: the family of H‐free graphs with minimum degree has bounded chromatic number. The study of was initiated in 1973 by Erd?s and Simonovits. Recently was determined for all graphs H . It is known that for all fixed , but that typically if Here we study the problem for sparse random graphs. We determine for most functions when , and also for all graphs H with © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 215–236, 2017     

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer , the binomial ‐uniform random hypergraph contains edge‐disjoint perfect matchings, provided , where is an integer depending only on . Our result for is asymptotically optimal and for is optimal up to the factor. This significantly improves a result of Frieze and Krivelevich.     

The random connection model: Connectivity,edge lengths,and degree distributions

Consider the random graph whose vertex set is a Poisson point process of intensity n on . Any two vertices are connected by an edge with probability , independently of all other edges, and independent of the other points of . d is the toroidal metric, r > 0 and is non‐increasing and . Under suitable conditions on g, almost surely, the critical parameter M_n for which does not have any isolated nodes satisfies . Let , and θ be the volume of the unit ball in . Then for all , is connected with probability approaching one as . The bound can be seen to be tight for the usual random geometric graph obtained by setting . We also prove some useful results on the asymptotic behavior of the length of the edges and the degree distribution in the connectivity regime. The results in this paper work for connection functions g that are not necessarily compactly supported but satisfy .     

Multipass greedy coloring of simple uniform hypergraphs

Let be the minimum number of edges in an n‐uniform simple hypergraph that is not two colorable. We prove that . Our result generalizes to r‐coloring of b‐simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b‐simple n‐uniform hypergraph that is not r‐colorable must be . By trimming arguments it implies that every such graph has edges. For any fixed our techniques yield also a lower bound for van der Waerden numbers W(n, r). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 125–146, 2016     

The stripping process can be slow: Part I

Given an integer k, we consider the parallel k‐stripping process applied to a hypergraph H: removing all vertices with degree less than k in each iteration until reaching the k‐core of H. Take H as : a random r‐uniform hypergraph on n vertices and m hyperedges with the uniform distribution. Fixing with , it has previously been proved that there is a constant such that for all m = cn with constant , with high probability, the parallel k‐stripping process takes iterations. In this paper, we investigate the critical case when . We show that the number of iterations that the process takes can go up to some power of n, as long as c approaches sufficiently fast. A second result we show involves the depth of a non‐k‐core vertex v: the minimum number of steps required to delete v from where in each step one vertex with degree less than k is removed. We will prove lower and upper bounds on the maximum depth over all non‐k‐core vertices.     

Spectra of large diluted but bushy random graphs

We compute an asymptotic expansion in of the limit in of the empirical spectral measure of the adjacency matrix of an Erd?s‐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., 49, 160–184, 2016     

Arboricity and spanning‐tree packing in random graphs

We study the arboricity and the maximum number of edge‐disjoint spanning trees of the classical random graph . For all , we show that, with high probability, is precisely the minimum of and , where is the minimum degree of the graph and denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for such that the following holds. Above this threshold, equals and equals . Below this threshold, equals , and we give a two‐value concentration result for the arboricity in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are randomly added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most in the two‐choice load balancing problem, where .     

A tail bound for read‐k families of functions

We prove a Chernoff‐like large deviation bound on the sum of non‐independent random variables that have the following dependence structure. The variables are arbitrary [0,1]‐valued functions of independent random variables , modulo a restriction that every X_i influences at most k of the variables . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 99–108, 2015     

Rainbow matchings and Hamilton cycles in random graphs

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