arXiv subjects

Asaf Ferber

Publications and source records attributed to Asaf Ferber.

At least 55 records · Page 3Linked to original sources

arXiv2017

Supersaturated sparse graphs and hypergraphs

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are $2^{(1+o(1))\text{ex}(n,H)}$ such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs $H$. We make a first attempt at addressing this enumeration problem for a general bipartite graph $H$. We show that an upper bound of $2^{O(\text{ex}(n,H))}$ on the number of $H$-free graphs with $n$ vertices follows merely from a rather natural assumption on the growth rate of $n \mapsto \text{ex}(n,H)$; an analogous statement remains true when $H$ is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdős and Simonovits. The bounds on the number of $H$-free hypergraphs are derived from it using the method of hypergraph containers.

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arXiv2017

Law of Iterated Logarithm for random graphs

A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$ $$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{σ_n \sqrt {2 \log \log n }} =1 \right] =1 . $$ In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph $H$. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random $k$-uniform hypergraphs, we obtain the Central Limit Theorem (CLT) and LIL for the number of Hamilton cycles.

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arXiv2017

Embedding large graphs into a random graph

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $Δ\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $Δ$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{-1}\log^{1/Δ}n)^{2/(Δ+1)}$. Our assumption on $p$ is optimal up to the $polylog$ factor. We note that this $polylog$ term matches the conjectured threshold for the spanning case.

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arXiv2017

Resilience for the Littlewood-Offord Problem

Consider the sum $X(ξ)=\sum_{i=1}^n a_iξ_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $ξ=(ξ_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\Pr[ξ_i=1]=\Pr[ξ_i=-1]=1/2$). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities $\Pr[X=x]$. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the $ξ_i$ is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present a few interesting open problems.

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arXiv2017

Spanning universality in random graphs

A graph is said to be $\mathcal{H}(n, Δ)$-universal if it contains every graph on $n$ vertices with maximum degree at most $Δ$. Using a `matching-based' embedding technique introduced by Alon and Füredi, Dellamonica, Kohayakawa, Rödl and Ruciński showed that the random graph $G_{n,p}$ is asymptotically almost surely $\mathcal{H}(n, Δ)$-universal for $p = \tilde Ω(n^{-1/Δ})$ - a threshold for the property that every subset of $Δ$ vertices has a common neighbour. This bound has become a benchmark in the field and many subsequent results on embedding spanning structures of maximum degree $Δ$ in random graphs are proven only up to this threshold. We take a step towards overcoming limitations of former techniques by showing that $G_{n,p}$ is almost surely $\mathcal{H}(n, Δ)$-universal for $p = \tilde Ω(n^{- 1/(Δ-1/2)})$.

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arXiv2016

Optimal Threshold for a Random Graph to be 2-Universal

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most two, $\mathcal{H}(n,2)$, we prove that there exists a constant $C$ such that for $p \geq C \left( \frac{\log n}{n^2} \right)^{\frac{1}{3}}$, the binomial random graph $G(n,p)$ is typically $\mathcal{H}(n,2)$-universal. This bound is optimal up to the constant factor as illustrated in the seminal work of Johansson, Kahn, and Vu for triangle factors. Our result improves significantly on the previous best bound of $p \geq C \left(\frac{\log n}{n}\right)^{\frac{1}{2}}$ due to Kim and Lee. In fact, we prove the stronger result that for the family of all graphs on $n$ vertices, of maximum degree at most two and of girth at least $\ell$, $\mathcal{H}^{\ell}(n,2)$, $G(n,p)$ is typically $\mathcal H^{\ell}(n,2)$-universal when $p \geq C \left(\frac{\log n}{n^{\ell -1}}\right)^{\frac{1}{\ell}}$. This result is also optimal up to the constant factor. Our results verify (in a weak form) a classical conjecture of Kahn and Kalai.

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arXiv2016

Counting Hamilton decompositions of oriented graphs

A Hamilton cycle in a directed graph $G$ is a cycle that passes through every vertex of $G$. A Hamiltonian decomposition of $G$ is a partition of its edge set into disjoint Hamilton cycles. In the late $60$s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus, who proved more generally that every $r$-regular $n$-vertex oriented graph $G$ (without antiparallel edges) with $r=cn$ for some fixed $c>3/8$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this paper we address the natural question of estimating the number of such decompositions of $G$ and show that this number is $n^{(1-o(1))cn^2}$. In addition, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

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arXiv2016

Finding Hamilton cycles in random graphs with few queries

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph possessing a given target property. We show that if $p\geq \frac{\ln n+\ln\ln n+ω(1)}{n}$, then one can find a Hamilton cycle with high probability after exposing $(1+o(1))n$ edges. Our result is tight in both $p$ and the number of exposed edges.

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arXiv2016

Finding paths in sparse random graphs requires many queries

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in $G\sim \mathcal G(n,p)$ when $p=\frac{1+\varepsilon}{n}$ for some fixed constant $\varepsilon>0$. This random graph is known to have typically linearly long paths. To have $\ell$ edges with high probability in $G\sim \mathcal G(n,p)$ one clearly needs to query at least $Ω\left(\frac{\ell}{p}\right)$ pairs of vertices. Can we find a path of length $\ell$ economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length $\ell=Ω\left(\frac{\log\left(\frac{1}{\varepsilon}\right)}{\varepsilon}\right)$ with at least constant probability in $G\sim \mathcal G(n,p)$ with $p=\frac{1+\varepsilon}{n}$ must query at least $Ω\left(\frac{\ell}{p\varepsilon \log\left(\frac{1}{\varepsilon}\right)}\right)$ pairs of vertices. This is tight up to the $\log\left(\frac{1}{\varepsilon}\right)$ factor.

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arXiv2016

Packing Loose Hamilton Cycles

A subset $C$ of edges in a $k$-uniform hypergraph $H$ is a \emph{loose Hamilton cycle} if $C$ covers all the vertices of $H$ and there exists a cyclic ordering of these vertices such that the edges in $C$ are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random $k$-uniform hypergraph $H^k_{n,p}$ has vertex set $[n]$ and an edge set $E$ obtained by adding each $k$-tuple $e\in \binom{[n]}{k}$ to $E$ with probability $p$, independently at random. Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but $o(|E|)$ edges, referred to as the \emph{packing problem}. While it is known that the threshold probability for the appearance of a loose Hamilton cycle in $H^k_{n,p}$ is $p=Θ\left(\frac{\log n}{n^{k-1}}\right)$, the best known bounds for the packing problem are around $p=\text{polylog}(n)/n$. Here we make substantial progress and prove the following asymptotically (up to a polylog$(n)$ factor) best possible result: For $p\geq \log^{C}n/n^{k-1}$, a random $k$-uniform hypergraph $H^k_{n,p}$ with high probability contains $N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$ edge-disjoint loose Hamilton cycles. Our proof utilizes and modifies the idea of "online sprinkling" recently introduced by Vu and the first author.

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arXiv2016

Packing trees of unbounded degrees in random graphs

In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ are $n$-vertex trees, each of which has maximum degree at most $(np)^{1/6} / (\log n)^6$. Then with high probability, one can find edge-disjoint copies of all the $T_i$ in the random graph $G_{n,p}$, provided that $p \geq (\log n)^{36}/n$ and $N \le (1-\varepsilon)np/2$ for a positive constant $\varepsilon$. Moreover, if each $T_i$ has at most $(1-α)n$ vertices, for some positive $α$, then the same result holds under the much weaker assumptions that $p \geq (\log n)^2/(cn)$ and $Δ(T_i) \leq c np / \log n$ for some~$c$ that depends only on $α$ and $\varepsilon$. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.

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arXiv2016

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.

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arXiv2016

Packing spanning graphs from separable families

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $ε$ and $Δ$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree at most $Δ$ such that $e(G_1)+\dotsb+e(G_t) \leq (1-ε)\binom{n}{2}$ packs into $K_n$. This improves results of Böttcher, Hladký, Piguet, and Taraz when $\mathcal G$ is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás (1976) for the case that all trees have maximum degree at most $Δ$. The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

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arXiv2016

Random matrices: Law of the iterated logarithm

The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated logarithm? We illustrate this possibility by showing that this is indeed the case for the log of the permanent of random Bernoulli matrices and pose open questions concerning several other matrix parameters.

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arXiv2016

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=ω(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.

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arXiv2016

Almost-spanning universality in random graphs

A graph $G$ is said to be $\mathcal H(n,Δ)$-universal if it contains every graph on $n$ vertices with maximum degree at most $Δ$. It is known that for any $\varepsilon > 0$ and any natural number $Δ$ there exists $c > 0$ such that the random graph $G(n,p)$ is asymptotically almost surely $\mathcal H((1-\varepsilon)n,Δ)$-universal for $p \geq c (\log n/n)^{1/Δ}$. Bypassing this natural boundary, we show that for $Δ\geq 3$ the same conclusion holds when $p = ω\left(n^{-\frac{1}{Δ-1}}\log^5 n\right)$.

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arXiv2015

Packing, Counting and Covering Hamilton cycles in random directed graphs

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability $p>\log^C(n)/n$. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

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arXiv2015

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ contains at least $(1-\varepsilon)\textrm{reg}_{even}(G)/2$ edge-disjoint Hamilton cycles, where $\reg(G)$ is the maximum \emph{even} degree of a spanning regular subgraph of $G$. This establishes an approximate version of a conjecture of Kühn, Lapinskas and Osthus.

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