arXiv subjects

Benny Sudakov

Publications and source records attributed to Benny Sudakov.

At least 73 records · Page 4Linked to original sources

arXiv2023

Optimal Hamilton covers and linear arboricity for random graphs

In his seminal 1976 paper, Pósa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of $G(n,p)$ with Hamilton cycles? How many cycles are necessary? In this paper we show that for $ p\geq C\log n/n$, we can cover $G\sim G(n,p)$ with precisely $\lceilΔ(G)/2\rceil$ Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability $p$, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szabó, and improves upon previous work of Hefetz, Kühn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.

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arXiv2023

On MaxCut and the Lovász theta function

In this short note we prove a lower bound for the MaxCut of a graph in terms of the Lovász theta function of its complement. We combine this with known bounds on the Lovász theta function of complements of $H$-free graphs to recover many known results on the MaxCut of $H$-free graphs. In particular, we give a new, very short proof of a conjecture of Alon, Krivelevich and Sudakov about the MaxCut of graphs with no cycles of length $r$.

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arXiv2023

Ascending Subgraph Decomposition

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph $G$ into copies $H_1, \ldots, H_m$ are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with $\binom{m+1}2$ edges can be decomposed into subgraphs $H_1, \dots, H_m$ such that each $H_i$ has $i$ edges and is isomorphic to a subgraph of $H_{i+1}$. In this paper we prove this conjecture for sufficiently large $m$.

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arXiv2023

The extremal number of cycles with all diagonals

In 1975, Erdős asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erdős observed that the upper bound $O(n^{5/3})$ holds since the complete bipartite graph $K_{3,3}$ can be viewed as a cycle of length six with all diagonals. In this paper, we resolve this old problem. We prove that there exists a constant $C$ such that every $n$-vertex with $Cn^{3/2}$ edges contains a cycle with all diagonals. Since any cycle with all diagonals contains cycles of length four, this bound is best possible using well-known constructions of graphs without a four-cycle based on finite geometry. Among other ideas, our proof involves a novel lemma about finding an `almost-spanning' robust expander which might be of independent interest.

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arXiv2023

Extremal, enumerative and probabilistic results on ordered hypergraph matchings

An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed and has connections and applications to extremal and enumerative combinatorics, probability, and geometry. On the other hand, in the case $r \ge 3$ much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered $r$-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain.

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arXiv2023

Ramsey numbers of hypergraphs of a given size

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

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arXiv2023

Ramsey problems for monotone paths in graphs and hypergraphs

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. We also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

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arXiv2023

Pancyclicity of Hamiltonian graphs

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most $k$ and at least $n = Ω(k^2)$ vertices is pancyclic. In this paper we prove this old conjecture in a strong form by showing that if such a graph has $n = (2+o(1))k^2$ vertices, it is already pancyclic, and this bound is asymptotically best possible.

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arXiv2023

Cycles with many chords

How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound by showing that $Ω(n\log^8n)$ edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.

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arXiv2023

Effective bounds for induced size-Ramsey numbers of cycles

The induced size-Ramsey number $\hat{r}_\text{ind}^k(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have such that for any $k$-coloring of its edges, there exists a monochromatic copy of $H$ which is an induced subgraph of $G$. In 1995, in their seminal paper, Haxell, Kohayakawa and Luczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., $\hat{r}_\text{ind}^k(C_n)\leq Cn$ for some $C=C(k)$. The constant $C$ comes from the use of the regularity lemma, and has a tower type dependence on $k$. In this paper we significantly improve these bounds, showing that $\hat{r}_\text{ind}^k(C_n)\leq O(k^{102})n$ when $n$ is even, thus obtaining only a polynomial dependence of $C$ on $k$. We also prove $\hat{r}_\text{ind}^k(C_n)\leq e^{O(k\log k)}n$ for odd $n$, which almost matches the lower bound of $e^{Ω(k)}n$. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies $\hat{r}^k(C_n)=e^{O(k)}n$ for odd $n$. This substantially improves the best previous result of $e^{O(k^2)}n$, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

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arXiv2023

On Ramsey size-linear graphs and related questions

In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erdős, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erdős, Faudree, Rousseau and Schelp. We prove the case $k=3$.

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arXiv2023

Maximal Chordal Subgraphs

A chordal graph is a graph with no induced cycles of length at least $4$. Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$. In the late '80s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$. In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$.

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arXiv2023

Hamilton cycles in pseudorandom graphs

Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well known conjecture in the area states that any $d$-regular $n$-vertex graph $G$ whose second largest eigenvalue in absolute value $λ(G)$ is at most $\frac{d}{C}$, for some universal constant $C>0$, has a Hamilton cycle. In this paper, we obtain two main results which make substantial progress towards this problem. Firstly, we settle this conjecture in full when the degree $d$ is at least a small power of $n$. Secondly, in the general case we show that $λ(G) \leq \frac{d}{C(\log n)^{1/3}}$ implies the existence of a Hamilton cycle, improving the 20-year old bound of $\frac{d}{ \log^{1-o(1)} n}$ of Krivelevich and Sudakov. We use in a novel way a variety of methods, such as a robust Pósa rotation-extension technique, the Friedman-Pippenger tree embedding with rollbacks and the absorbing method, combined with additional tools and ideas. Our results have several interesting applications, giving best bounds on the number of generators which guarantee the Hamiltonicity of random Cayley graphs, which is an important partial case of the well known Hamiltonicity conjecture of Lovász. They can also be used to improve a result of Alon and Bourgain on additive patterns in multiplicative subgroups.

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arXiv2023

A generalization of Bondy's pancyclicity theorem

The bipartite independence number of a graph $G$, denoted as $\tildeα(G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $δ(G)\geq\tilde α(G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $δ(G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac's Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$. In this paper we show that $δ(G)\geq\tilde α(G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

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arXiv2023

Tight Ramsey bounds for multiple copies of a graph

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs $G$ for which the value of $r(G)$ is known exactly. One such family consists of large vertex disjoint unions of a fixed graph $H$, we denote such a graph, consisting of $n$ copies of $H$ by $nH$. This classical result was proved by Burr, Erdős and Spencer in 1975, who showed $r(nH)=(2|H|-α(H))n+c$, for some $c=c(H)$, provided $n$ is large enough. Since it did not follow from their arguments, Burr, Erdős and Spencer further asked to determine the number of copies we need to take in order to see this long term behaviour and the value of $c$. More than $30$ years ago Burr gave a way of determining $c(H)$, which only applies when the number of copies $n$ is triple exponential in $|H|$. In this paper we give an essentially tight answer to this very old problem of Burr, Erdős and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.

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arXiv2023

The Minimum Degree Removal Lemma Thresholds

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains $δn^{v(H)}$ copies of $H$ for some $δ= δ(\varepsilon,H) > 0$. The current proofs of the removal lemma give only very weak bounds on $δ(\varepsilon,H)$, and it is also known that $δ(\varepsilon,H)$ is not polynomial in $\varepsilon$ unless $H$ is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that $δ(\varepsilon,H)$ depends polynomially or linearly on $\varepsilon$. In this paper we answer several questions of Fox and Wigderson on this topic.

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arXiv2023

Chvátal-Erdős condition for pancyclicity

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $κ(G) > (1+o(1)) α(G)$ is pancyclic. This extends the famous Chvátal-Erdős condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.

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arXiv2023

Large independent sets from local considerations

The following natural problem was raised independently by Erdős-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $α(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of size $r$? In this paper we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve previously best lower bounds due to Linial-Rabinovich, Erdős-Hajnal and Alon-Sudakov. As an example, we prove that any $n$-vertex graph $G$ having an independent set of size $3$ among every $7$ vertices has $α(G) \ge Ω(n^{5/12})$. This confirms a conjecture of Erdős and Hajnal that $α(G)$ should be at least $n^{1/3+\varepsilon}$ and brings the exponent half-way to the best possible value of $1/2$. Our second approach deals with upper bounds. It relies on a reduction of the original question to the following natural extremal problem. What is the minimum possible value of the $2$-density of a graph on $m$ vertices having no independent set of size $r$? This allows us to improve previous upper bounds due to Linial-Rabinovich, Krivelevich and Kostochka-Jancey. As part of our arguments we link the problem of Erdős-Hajnal and Linial-Rabinovich and our new extremal $2$-density problem to a number of other well-studied questions. This leads to many interesting directions for future research.

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