Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

List colorings of multipartite hypergraphs

Let χ_l(G) denote the list chromatic number of the r‐uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d‐regular, then . To see how close this inequality is to best possible, we examine χ_l(G) when G is a random r‐partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that almost surely, where d is the expected average degree of G and . The function g(r,α) is defined in terms of “preference orders” and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on χ_l(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4.     

Finding a Hamilton cycle fast on average using rotations and extensions

We present an algorithm CRE, which either finds a Hamilton cycle in a graph G or determines that there is no such cycle in the graph. The algorithm's expected running time over input distribution G～G(n,p) is (1+o(1))n/p, the optimal possible expected time, for . This improves upon previous results on this problem due to Gurevich and Shelah, and to Thomason.     

On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.     

Fractional clique decompositions of dense graphs

For each , we show that any graph G with minimum degree at least has a fractional K_r‐decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional K_r‐decomposition given by Dukes (for small r) and Barber, Kühn, Lo, Montgomery, and Osthus (for large r), giving the first bound that is tight up to the constant multiple of r (seen, for example, by considering Turán graphs). In combination with work by Glock, Kühn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic number , and any , any sufficiently large graph G with minimum degree at least has, subject to some further simple necessary divisibility conditions, an (exact) F‐decomposition.     

On giant components and treewidth in the layers model

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 ?‐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 T₃(G) has a connected component of size , and moreover, T₃(G) has treewidth . In contrast, T₂(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 T₂(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     

Tree decompositions of graphs without large bipartite holes

A recent result of Condon, Kim, Kühn, and Osthus implies that for any , an n‐vertex almost r‐regular graph G has an approximate decomposition into any collections of n‐vertex bounded degree trees. In this paper, we prove that a similar result holds for an almost αn‐regular graph G with any α>0 and a collection of bounded degree trees on at most (1?o(1))n vertices if G does not contain large bipartite holes. This result is sharp in the sense that it is necessary to exclude large bipartite holes and we cannot hope for an approximate decomposition into n‐vertex trees. Moreover, this implies that for any α>0 and an n‐vertex almost αn‐regular graph G, with high probability, the randomly perturbed graph has an approximate decomposition into all collections of bounded degree trees of size at most (1?o(1))n simultaneously. This is the first result considering an approximate decomposition problem in the context of Ramsey‐Turán theory and the randomly perturbed graph model.     

Spanning trees in randomly perturbed graphs

A classical result of Komlós, Sárközy, and Szemerédi states that every n‐vertex graph with minimum degree at least (1/2 + o(1))n contains every n‐vertex tree with maximum degree . Krivelevich, Kwan, and Sudakov proved that for every n‐vertex graph G_α with minimum degree at least αn for any fixed α > 0 and every n‐vertex tree T with bounded maximum degree, one can still find a copy of T in G_α with high probability after adding O(n) randomly chosen edges to G_α. We extend the latter results to trees with (essentially) unbounded maximum degree; for a given and α > 0, we determine up to a constant factor the number of random edges that we need to add to an arbitrary n‐vertex graph with minimum degree αn in order to guarantee with high probability a copy of any fixed n‐vertex tree with maximum degree at most Δ.     

An extension of the Erdős‐Tetali theorem

Given a sequence , let r_??,h(n) denote the number of ways n can be written as the sum of h elements of ??. Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O‐regularly varying and of positive increase) satisfying then there must exist with for which r_{??,h + ?}(n) = Θ(f(n)^{h + ?}/n) for all ? ≥ 0. Furthermore, for h = 2 this condition can be weakened to . The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.     

On the maximal multiplicity of block sizes in a random set partition

We study the asymptotic behavior of the maximal multiplicity M_n = M_n(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with We^W = n. We show that, over subsequences {n_k}_{k ≥ 1} of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z₁ and Z₂ are independent copies of a standard normal random variable. The subsequences {n_k}_{k ≥ 1}, where the weak convergence is observed, and the quantity u depend on the fractional part f_n of the function W(n). In particular, we establish that . The behavior of the largest multiplicity M_n is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.     

Graph limits of random graphs from a subset of connected k‐trees

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree G_n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that G_n,k, scaled by where H_k is the kth harmonic number and σ_Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G_∞,k.     

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 ? o(1))np Hamilton cycles C₁,…,C_t, each of which is oriented arbitrarily, a digraph ～(n, p) w.h.p. contains edge disjoint copies of C₁,…,C_t, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ～(n, p) w.h.p. contains (1 ± o(1))n!pⁿ copies of C, provided .     

Coloring graphs from random lists of fixed size

Let G = G(n) be a graph on n vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as ) of the existence of a proper coloring ? of G, such that for every vertex v of G. We show, for all fixed k and growing n, that if , then the probability that G has such a proper coloring tends to 1 as . A similar result for complete graphs is also obtained: if and L is a random ‐list assignment for the complete graph K_n on n vertices, then the probability that K_n has a proper coloring with colors from the random lists tends to 1 as .Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317‐327, 2014     

Mantel's theorem for random hypergraphs

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 F₅, 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 F₅ is tripartite for n > 3000. A natural question is for what p is every maximum F₅‐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., 48, 641–654, 2016     

When does the K4‐free process stop?

The K₄‐free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K₄. Let G be the random maximal K₄‐free graph obtained at the end of the process. We show that for some positive constant C, with high probability as , the maximum degree in G is at most . This resolves a conjecture of Bohman and Keevash for the K₄‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has edges and is ‘nearly regular’, i.e., every vertex has degree . This answers a question of Erd?s, Suen and Winkler for the K₄‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least , which matches an upper bound obtained by Bohman up to a factor of . Our analysis of the K₄‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erd?s and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014     

Testing for forbidden order patterns in an array

A sequence contains a pattern , that is, a permutations of [k], iff there are indices i₁ < … < i_k, such that f(i_x) > f(i_y) whenever π(x) > π(y). Otherwise, f is π‐free. We study the property testing problem of distinguishing, for a fixed π, between π‐free sequences and the sequences which differ from any π‐free sequence in more than ? n places. Our main findings are as follows: (1) For monotone patterns, that is, π = (k,k ? 1,…,1) and π = (1,2,…,k), there exists a nonadaptive one‐sided error ?‐test of query complexity. For any other π, any nonadaptive one‐sided error test requires queries. The latter lower‐bound is tight for π = (1,3,2). For specific it can be strengthened to Ω(n^{1 ? 2/(k + 1)}). The general case upper‐bound is O(?^?1/kn^{1 ? 1/k}). (2) For adaptive testing the situation is quite different. In particular, for any there exists an adaptive ?‐tester of query complexity.     

Edge‐coloring linear hypergraphs with medium‐sized edges

Motivated by the Erdos?‐Faber‐Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper‐edge sizes are bounded between i and inclusive, then there is a list edge coloring using colors. The dependence on n in the upper bound is optimal (up to the value of C_i,?).     

Sharp asymptotic for the chemical distance in long‐range percolation

We consider instances of long‐range percolation on and , where points at distance r get connected by an edge with probability proportional to r^?s, for s ∈ (d,2d), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distance D(x,y) between x and y in the limit as |x ? y|→∞. For the model on we show that, in probability as |x|→∞, the distance D(0,x) is squeezed between two positive multiples of , where for γ: = s/(2d). For the model on we show that D(0,xr) is, in probability as r→∞ for any nonzero , asymptotic to for φ a positive, continuous (deterministic) function obeying φ(r^γ) = φ(r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly‐exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.     

Spatial mixing and nonlocal Markov chains

We consider spin systems with nearest‐neighbor interactions on an n‐vertex d‐dimensional cube of the integer lattice graph . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O(1) bound for the relaxation time of the Swendsen‐Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is . Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.