Hypergraph coloring up to condensation

Improving a result of Dyer, Frieze and Greenhill [Journal of Combinatorial Theory, Series B, 2015], we determine the q‐colorability threshold in random k‐uniform hypergraphs up to an additive error of , where . The new lower bound on the threshold matches the “condensation phase transition” predicted by statistical physics considerations [Krzakala et al., PNAS 2007].     

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,?).     

Semi‐random graph process

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties , we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies . Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.     

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.     

Rapid mixing of hypergraph independent sets

We prove that the mixing time of the Glauber dynamics for sampling independent sets on n‐vertex k‐uniform hypergraphs is when the maximum degree Δ satisfies Δ ≤ c2^k/2, improving on the previous bound Bordewich and co‐workers of Δ ≤ k ? 2. This result brings the algorithmic bound to within a constant factor of the hardness bound of Bezakova and co‐workers which showed that it is NP‐hard to approximately count independent sets on hypergraphs when Δ ≥ 5·2^k/2.     

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 .     

On the weight distribution of random binary linear codes

We investigate the weight distribution of random binary linear codes. For 0 < λ < 1 and n→∞ pick uniformly at random λn vectors in and let be the orthogonal complement of their span. Given 0 < γ < 1/2 with 0 < λ < h(γ) let X be the random variable that counts the number of words in C of Hamming weight γn. In this paper we determine the asymptotics of the moments of X of all orders .     

Vanishing of cohomology groups of random simplicial complexes

We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.     

Perfect matchings and Hamiltonian cycles in the preferential attachment model

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if , then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that . One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are “older” (ie, are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting—sometimes called uniform attachment—in which vertices are added one by one and each vertex connects to m older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.     

Planar graphs: Random walks and bipartiteness testing

We initiates the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of for graphs on n vertices. The constant‐time testability was only known for planar graphs with bounded degree. Our algorithm is based on random walks. Since planar graphs have good separators, that is, bad expansion, our analysis diverges from standard techniques that involve the fast convergence of random walks on expanders. We reduce the problem to the task of detecting an odd‐parity cycle in a multigraph induced by constant‐length cycles. We iteratively reduce the length of cycles while preserving the detection probability, until the multigraph collapses to a collection of easily discoverable self‐loops. Our approach extends to arbitrary minor‐free graphs. We also believe that our techniques will find applications to testing other properties in arbitrary minor‐free graphs.     

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].     

On the discrepancy of random matrices with many columns

Motivated by the Komlós conjecture in combinatorial discrepancy, we study the discrepancy of random matrices with m rows and n independent columns drawn from a bounded lattice random variable. We prove that for n at least polynomial in m, with high probability the ?_∞‐discrepancy is at most twice the ?_∞‐covering radius of the integer span of the support of the random variable. Applying this result to random t‐sparse matrices, that is, uniformly random matrices with t ones and m?t zeroes in each column, we show that the ?_∞‐discrepancy is at most 2 with probability for . This improves on a bound proved by Ezra and Lovett showing the same bound for n at least m^t.     

Powers of tight Hamilton cycles in randomly perturbed hypergraphs

For k ≥ 2 and r ≥ 1 such that k + r ≥ 4, we prove that, for any α > 0, there exists ε > 0 such that the union of an n‐vertex k‐graph with minimum codegree and a binomial random k‐graph with on the same vertex set contains the rth power of a tight Hamilton cycle with high probability. This result for r = 1 was first proved by McDowell and Mycroft.     

Monochromatic paths in random tournaments

We prove that, with high probability, any 2‐edge‐coloring of a random tournament on n vertices contains a monochromatic path of length . This resolves a conjecture of Ben‐Eliezer, Krivelevich, and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.     

Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

We show that for every there exists C > 0 such that if then asymptotically almost surely the random graph contains the k^th power of a Hamilton cycle. This determines the threshold for appearance of the square of a Hamilton cycle up to the logarithmic factor, improving a result of Kühn and Osthus. Moreover, our proof provides a randomized quasi‐polynomial algorithm for finding such powers of cycles. Using similar ideas, we also give a randomized quasi‐polynomial algorithm for finding a tight Hamilton cycle in the random k‐uniform hypergraph for . The proofs are based on the absorbing method and follow the strategy of Kühn and Osthus, and Allen et al. The new ingredient is a general Connecting Lemma which allows us to connect tuples of vertices using arbitrary structures at a nearly optimal value of p. Both the Connecting Lemma and its proof, which is based on Janson's inequality and a greedy embedding strategy, might be of independent interest.     

A periodic isoperimetric problem related to the unique games conjecture

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space satisfies ?Ω = Ω^c and Ω + v = Ω^c (up to measure zero sets) for every standard basis vector . For any and for any q ≥ 1, let and let . For any x ∈ ?Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖₂ = 1. Let . Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: In particular, Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10^?9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.     

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

Information percolation and cutoff for the random‐cluster model

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017