On the cycle space of a random graph

Write for the cycle space of a graph G, for the subspace of spanned by the copies of the κ‐cycle in G, for the class of graphs satisfying , and for the class of graphs each of whose edges lies in a . We prove that for every odd and , so the 's of a random graph span its cycle space as soon as they cover its edges. For κ = 3 this was shown in [6].     

Phase transitions in graphs on orientable surfaces

Let be the orientable surface of genus and denote by the class of all graphs on vertex set with edges embeddable on . We prove that the component structure of a graph chosen uniformly at random from features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd?s‐Rényi random graph chosen uniformly at random from all graphs with vertex set and edges. It takes place at , when the giant component emerges. The second phase transition occurs at , when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from and has only been observed for graphs on surfaces.     

On polynomial approximations to AC

Classical approximation results show that any circuit of size and depth has an ‐error probabilistic polynomial over the reals of degree . We improve this upper bound to , which is much better for small values of . We then use this result to show that ‐wise independence fools circuits of size and depth up to error at most , improving on Tal's strengthening of Braverman's result that ‐wise independence suffices. To our knowledge, this is the first PRG construction for that achieves optimal dependence on the error . We also prove lower bounds on the best polynomial approximations to . We show that any polynomial approximating the function on bits to a small constant error must have degree at least . This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016).     

Swendsen‐Wang algorithm on the mean‐field Potts model

We study the q‐state ferromagnetic Potts model on the n‐vertex complete graph known as the mean‐field (Curie‐Weiss) model. We analyze the Swendsen‐Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single‐site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen‐Wang algorithm for the mean‐field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) for , (ii) for , (iii) for , where β_c is the critical temperature for the ordered/disordered phase transition. In contrast, for there are two critical temperatures that are relevant. We prove that the mixing time of the Swendsen‐Wang algorithm for the ferromagnetic Potts model on the n‐vertex complete graph satisfies: (i) for , (ii) for , (iii) for , and (iv) for . These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.     

The height of depth‐weighted random recursive trees

In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex . In this model, the probability of choosing depends on its depth in the tree. In particular, we assume that there is a function such that if has depth then its probability of being chosen is proportional to . We consider the expected value of the diameter of this model as determined by , and for various increasing we find expectations that range from polylogarithmic to linear.     

A scaling limit for the length of the longest cycle in a sparse random digraph

We discuss the length of the longest directed cycle in the sparse random digraph , constant. We show that for large there exists a function such that a.s. The function where is a polynomial in . We are only able to explicitly give the values , although we could in principle compute any .     

Rainbow structures in locally bounded colorings of graphs

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally ‐bounded if every vertex is incident to at most edges of each color, and is (globally) ‐bounded if every color appears at most times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored into rainbow almost‐spanning cycles; (2) approximate decompositions of edge‐colored into rainbow Hamilton cycles, provided that the coloring is ‐bounded and locally ‐bounded; and (3) an approximate decomposition into full transversals of any array, provided each symbol appears times in total and only times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow ‐factors, where is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi‐Hollingsworth conjecture on decompositions into rainbow spanning trees.     

On percolation and ‐hardness

The edge‐percolation and vertex‐percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst‐case instances. Specifically, we show that a number of classical ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming ). We also prove hardness results for other ‐hard problems such as Constraint Satisfaction Problems and Subset‐Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number and the chromatic number are robust to percolation in the following sense. Given a graph G, let be the graph obtained by randomly deleting edges of G with some probability . We show that if is small, then remains small with probability at least 0.99. Similarly, we show that if is large, then remains large with probability at least 0.99. We believe these results are of independent interest.     

A characterization of constant‐sample testable properties

We characterize the set of properties of Boolean‐valued functions on a finite domain that are testable with a constant number of samples (x,f(x)) with x drawn uniformly at random from . Specifically, we show that a property is testable with a constant number of samples if and only if it is (essentially) a k‐part symmetric property for some constant k, where a property is k‐part symmetric if there is a partition of such that whether satisfies the property is determined solely by the densities of f on . We use this characterization to show that symmetric properties are essentially the only graph properties and affine‐invariant properties that are testable with a constant number of samples and that for every constant , monotonicity of functions on the d‐dimensional hypergrid is testable with a constant number of samples.     

Asymptotics in percolation on high‐girth expanders

We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in the previous study that the second largest component, at any , has size at most for some . We show that, unlike the situation in the classical Erd?s‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size for arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, for example, the existence of a linear path.     

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 .     

A complete solution to existence of H designs

An H design is a triple , where is a set of points, a partition of into disjoint sets of size , and a set of ‐element transverses of , such that each ‐element transverse of is contained in exactly one of them. In 1990, Mills determined the existence of an H design with . In this paper, an efficient construction shows that an H exists for any integer with . Consequently, the necessary and sufficient conditions for the existence of an H design are , , and , with a definite exception .     

On the honeymoon Oberwolfach problem

The honeymoon Oberwolfach problem HOP asks the following question. Given newlywed couples at a conference and round tables of sizes , is it possible to arrange the participants at these tables for meals so that each participant sits next to their spouse at every meal and sits next to every other participant exactly once? A solution to HOP is a decomposition of , the complete graph with additional copies of a fixed 1‐factor , into 2‐factors, each consisting of disjoint ‐alternating cycles of lengths . It is also equivalent to a semi‐uniform 1‐factorization of of type ; that is, a 1‐factorization such that for all , the 2‐factor consists of disjoint cycles of lengths . In this paper, we first introduce the honeymoon Oberwolfach problem and then present several results. Most notably, we completely solve the case with uniform cycle lengths, that is, HOP. In addition, we show that HOP has a solution in each of the following cases: ; is odd and ; as well as for all . We also show that HOP has a solution whenever is odd and the Oberwolfach problem with tables of sizes has a solution.     

Shotgun reconstruction in the hypercube

Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the -dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for , almost every -coloring can be reconstructed from the multiset of colorings of 1-balls.     

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck‐Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an discrepancy bound when n ≤ m and an O(1) bound when n ? m^t. In this paper, we give a tight bound for the entire range of n and m, under a mild assumption that . The result is based on two steps. First, applying the partial coloring method to the case when and using the properties of the random set system we show that the overall discrepancy incurred is at most . Second, we reduce the general case to that of using LP duality and a careful counting argument.     

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.     

Dimension of the space generated by Steiner systems

To every Steiner system with parameters on and blocks , we can assign its characteristic vector , which is a ‐vector whose entries are indexed by the ‐subsets of such that for each ‐subset of if and only if . In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters is , provided that and there is at least one such system.     

High nonassociativity in order 8 and an associative index estimate

Let be a quasigroup. Put and assume that . Let and be the number of left and right translations of that are fixed point free. Put . Denote by the number of idempotents of . It is shown that . Call extremely nonassociative if . The paper reports what seems to be the first known example of such a quasigroup, with , , and . It also provides supporting theory for a search that verified for all quasigroups of order .     

The size-Ramsey number of powers of paths

Given graphs and and a positive integer , say that is -Ramsey for , denoted , if every -coloring of the edges of contains a monochromatic copy of . The size-Ramsey number of a graph is defined to be . Answering a question of Conlon, we prove that, for every fixed , we have , where is the th power of the -vertex path (ie, the graph with vertex set and all edges such that the distance between and in is at most ). Our proof is probabilistic, but can also be made constructive.