Papers by Benjamin Sudakov
Journal of Graph Theory, 2005
In this paper we obtain an asymptotic generalization of Turán's theorem. We prove that if all the... more In this paper we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non-trivial eigenvalues of a d-regular graph G on n vertices are sufficiently small, then the largest K t-free subgraph of G contains approximately t−2 t−1-fraction of its edges. Turán's theorem corresponds to the case d = n − 1.
Combinatorics, Probability and Computing, 2012
Consider the random graph process where we start with an empty graph on n vertices and, at time t... more Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting g...
Proceedings of the American Mathematical Society, 2020
We study the following question raised by Erdős and Hajnal in the early 90's. Over all n-vertex g... more We study the following question raised by Erdős and Hajnal in the early 90's. Over all n-vertex graphs G what is the smallest possible value of m for which any m vertices of G contain both a clique and an independent set of size log n? We construct examples showing that m is at most 2 2 (log log n) 1/2+o(1) obtaining a twofold sub-polynomial improvement over the upper bound of about √ n coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log n in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.
The Electronic Journal of Combinatorics, 2001
Let $T$ be a tree with $t$ vertices. Clearly, an $n$ vertex graph contains at most $n/t$ vertex d... more Let $T$ be a tree with $t$ vertices. Clearly, an $n$ vertex graph contains at most $n/t$ vertex disjoint trees isomorphic to $T$. In this paper we show that for every $\epsilon >0$, there exists a $D(\epsilon,t)>0$ such that, if $d>D(\epsilon,t)$ and $G$ is a simple $d$-regular graph on $n$ vertices, then $G$ contains at least $(1-\epsilon)n/t$ vertex disjoint trees isomorphic to $T$.
The Electronic Journal of Combinatorics, 2001
The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of ord... more The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.
Cpc, 2007
For a fixed graph H, we define the rainbow Turán number ex * (n, H) to be the maximum number of e... more For a fixed graph H, we define the rainbow Turán number ex * (n, H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n, H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex * (n, H) = (1+o(1))ex(n, H), and if H is colour-critical we show that ex * (n, H) = ex(n, H). When H is the complete bipartite graph K s,t with s ≤ t we show ex * (n, K s,t) = O(n 2−1/s), which matches the known bounds for ex(n, K s,t) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex * (n, C 6) = O(n 4/3), which is of the correct order of magnitude.
The Ramsey number r(G) of a graph G is the minimum N such that every red-blue coloring of the edg... more The Ramsey number r(G) of a graph G is the minimum N such that every red-blue coloring of the edges of the complete graph on N vertices contains a monochromatic copy of G. Determining or estimating these numbers is one of the central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erdős and Szekeres in 1935, asserts that the Ramsey number of the complete graph with m edges is at most 2 O(√ m). Motivated by this estimate Erdős conjectured, more than a quarter century ago, that there is an absolute constant c such that r(G) ≤ 2 c √ m for any graph G with m edges and no isolated vertices. In this short note we prove this conjecture.
Combinatorics, Probability and Computing, 2014
Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can b... more Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G(n, p) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G(n, p) can be decomposed into a union of n/4 + np/2 + o(n) cycles and edges w.h.p. This result is asymptotically tight.
Bolyai Society Mathematical Studies, 2006
SIAM Journal on Computing, 2004
We consider the problem of learning a matching (i.e., a graph in which all vertices have degree 0... more We consider the problem of learning a matching (i.e., a graph in which all vertices have degree 0 or 1) in a model where the only allowed operation is to query whether a set of vertices induces an edge. This is motivated by a problem that arises in molecular biology. In the deterministic nonadaptive setting, we prove a (1 2 + o(1)) n 2 upper bound and a nearly matching 0.32 n 2 lower bound for the minimum possible number of queries. In contrast, if we allow randomness, then we obtain (by a randomized, nonadaptive algorithm) a much lower O(n log n) upper bound, which is best possible (even for randomized fully adaptive algorithms).
Random Structures and Algorithms, 2001
Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to... more Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d = d n grows more quickly than √ n. These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others.
Random Structures and Algorithms, 2006
We study a model of random graphs, where a random instance is obtained by adding random edges to ... more We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman et al. in [3], [4]. Here we obtain a sharp threshold for the appearance of a fixed subgraph, and for certain Ramsey properties. We also consider a related model of random k-SAT formulas, where an instance is obtained by adding random k-clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non-satisfiability of thus obtained random formula.
Journal of the European Mathematical Society, 2013
We describe two constructions of (very) dense graphs which are edge disjoint unions of large indu... more We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with N 2 − o(N 2) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N 1−o(1). The second construction provides a covering of all edges of the complete graph K N by two graphs, each being the edge disjoint union of at most N 2−δ induced matchings, where δ > 0.076. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of Håstad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.
Journal of Graph Theory, 2003
A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G... more A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdős and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n δ(C). Our methods enable us also to show that under much weaker assumption, i.e., q(G) ≤ n/14, G still must contain an induced subgraph with exactly y edges, for every y between 0 and e Ω(√ log n) .
Journal of Graph Theory, 2007
Let G be a graph on n vertices in which every induced subgraph on s = log 3 n vertices has an ind... more Let G be a graph on n vertices in which every induced subgraph on s = log 3 n vertices has an independent set of size at least t = log n. What is the largest q = q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdős and Hajnal. We show that q(n) = Θ(log 2 n/ log log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.
Journal of Combinatorial Theory, Series B, 2007
Let G be a graph with maximum degree ∆ whose vertex set is partitioned into parts V (G) = V 1 ∪. ... more Let G be a graph with maximum degree ∆ whose vertex set is partitioned into parts V (G) = V 1 ∪. . .∪ V r. A transversal is a subset of V (G) containing exactly one vertex from each part V i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V i , taken over all choices of V i and v ∈ V i. We prove that for every fixed ǫ > 0, if all part sizes |V i | ≥ (1 + ǫ)∆ and the local degree of G is o(∆), then G has an independent transversal for sufficiently large ∆. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s = 2.) In that context, we prove that parts of size |V i | ≥ (1 + ǫ) ∆ s−1 and local degree o(∆) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.
Journal of Combinatorial Theory, Series B, 2012
More than twenty years ago, Manickam, Miklós, and Singhi conjectured that for any integers n, k s... more More than twenty years ago, Manickam, Miklós, and Singhi conjectured that for any integers n, k satisfying n ≥ 4k, every set of n real numbers with nonnegative sum has at least n−1 k−1 kelement subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for n ≥ 33k 2. This substantially improves the best previously known exponential lower bound n ≥ e ck log log k. In addition we prove a tight stability result showing that for every k and all sufficiently large n, every set of n reals with a nonnegative sum that does not contain a member whose sum with any other k − 1 members is nonnegative, contains at least n−1 k−1 + n−k−1 k−1 − 1 subsets of cardinality k with nonnegative sum.
Israel Journal of Mathematics, 2006
In this paper we study non-interactive correlation distillation (NICD), a generalization of noise... more In this paper we study non-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following:
Geometric and Functional Analysis, 2009
e(X) − p |X| 2 2 ≤ β|X| for every subset X ⊆ V , where e(X) stands for the number of edges spanne... more e(X) − p |X| 2 2 ≤ β|X| for every subset X ⊆ V , where e(X) stands for the number of edges spanned by X in G. Informally, this definition indicates that the edge distribution of G is similar to that of the random graph G |V |,p , where the degree of similarity is controlled by parameter β. Here are the main results of this paper. Theorem 1 Let 0 < α < 1 be a constant. Let G be a (t, α)-expanding graph of order n, and let t ≥ 10. Then G contains a minor with average degree at least c √ nt log t √ log n , where c = c(α) > 0 is a constant. This is an extension of results of Alon, Seymour and Thomas [5], Plotkin, Rao and Smith [33], and of Kleinberg and Rubinfeld [16], who cover basically the case of expansion by a constant factor t = Θ(1). Theorem 2 Let G be a (p, β)-jumbled graph of order n such that β = o(np). Then G contains a minor with average degree cn √ p, for an absolute constant c > 0. This statement is an extension of results of A. Thomason [39, 40], who studied the case of constant p. It can be also used to derive some of the results of Drier and Linial [12]. Theorem 3 Let 2 ≤ s ≤ s ′ be integers. Let G be a K s,s ′-free graph with average degree r. Then G contains a minor with average degree cr 1+ 1 2(s−1) , where c = c(s, s ′) > 0 is a constant. This confirms a conjecture of Kühn and Osthus from [21]. Theorem 4 Let k ≥ 2 and let G be a C 2k-free graph with average degree r. Then G contains a minor with average degree cr k+1 2 , where c = c(k) > 0 is a constant. This theorem generalizes results of Thomassen [37], Diestel and Rompel [11], and Kühn and Osthus [22], who proved similar statements under the (much more restrictive) assumption that G has girth at least 2k + 1. All of the above results are, up to a constant factor, asymptotically tight (Theorems 1, 2), or are allegedly tight (Theorems 3, 4), where in the latter case the tightness hinges upon widely accepted conjectures from Extremal Graph Theory about the asymptotic behavior of the Turán numbers of K s,s ′ and of C 2k .
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Papers by Benjamin Sudakov