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Mathias Schacht

Publications and source records attributed to Mathias Schacht.

At least 37 records · Page 2Linked to original sources
arXiv2019

Embedding spanning subgraphs in uniformly dense and inseparable graphs

We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $μn$ for arbitrarily small $μ>0$. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $μ=\frac{k}{k+1}$ suffices for large $n$. For smaller values of $μ$ the given graph $G$ must satisfy additional assumptions. We show that inducing subgraphs of density $d>0$ on linear subsets of vertices and being inseparable, in the sense that every cut has density at least $μ>0$, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalises recent results of Staden and Treglown.

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arXiv2019

Hamiltonian Berge cycles in random hypergraphs

In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős--Rényi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for \textit{weak Berge} cycles as well, thus resolving a conjecture of Poole.

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arXiv2019

Odd cycles in subgraphs of sparse pseudorandom graphs

We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $Γ$ be an $(n,d,λ)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-λ,λ]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $λ^{2k-1}\ll d^{2k}/n$, every subgraph $G$ of $Γ$ with $(1/2+o(1))e(Γ)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, Hàn, and the third author proved a weaker statement by allowing an extra polylogarithmic factor in the assumption $λ^{2k-1}\ll d^{2k}/n$, but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabo, and Vu's Turán-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of $Γ$ contains at least $(1-o(1))2^{-2k}d^{2k+1}$ monochromatic copies of $C_{2k+1}$. Both results are asymptotically best possible by Alon and Kahale's construction of $C_{2k+1}$-free pseudorandom graphs.

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arXiv2019

Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphs

For positive integers $r > \ell$, an $r$-uniform hypergraph is called an $\ell$-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of $r$ consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely $\ell$ vertices. Such cycles are said to be linear when $\ell = 1$, and nonlinear when $\ell > 1$. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all $r > \ell > 1$, the threshold $p^*_{r, \ell} (n)$ for the appearance of a Hamiltonian $\ell$-cycle in the random $r$-uniform hypergraph on $n$ vertices is sharp and is $p^*_{r, \ell} (n) = λ(r,\ell) (\frac{\mathrm{e}}{n})^{r - \ell}$ for an explicitly specified function $λ$. This resolves several questions raised by Dudek and Frieze in 2011.

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arXiv2019

Powers of Hamiltonian cycles in randomly augmented graphs

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every $k\geq 1$ and sufficiently large $n$ already the minimum degree $δ(G)\ge\tfrac{k}{k+1}n$ for an $n$-vertex graph $G$ alone suffices to ensure the existence of a $k$-th power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just $O(n)$ random edges ensures the presence of the $(k+1)$-st power of a Hamiltonian cycle with probability close to one.

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arXiv2019

Forcing quasirandomness with triangles

We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families $\\mathcal F$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in(0,1]$ for every $F\in \mathcal F$, then $G$ is quasirandom with density $p$. Such families $\mathcal F$ are said to be forcing. Several forcing families were found over the last three decades and characterising all bipartite graphs $F$ such that $(K_2,F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko's conjecture. In fact, most of the known forcing families involve bipartite graphs only. We consider forcing pairs containing the triangle $K_3$. In particular, we show that if $(K_2,F)$ is a forcing pair, then so is $(K_3,F')$, where $F'$ is obtained from $F$ by replacing every edge of $F$ by a triangle (each of which introduces a new vertex). For the proof we first show that $(K_3,C'_4)$ is a forcing pair, which strengthens related results of Simonovits and Sós and of Conlon et al.

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arXiv2019

On the local density problem for graphs of given odd-girth

Erdős conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andrásfai graphs. As a consequence, Erdős' conjecture holds for every triangle-free graph $G$ with minimum degree $δ(G)>10n/29$ and if $χ(G)\leq 3$ the degree condition can be relaxed to $δ(G)> n/3$. In fact, we obtain a more general result for graphs of higher odd-girth.

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arXiv2019

Homomorphism thresholds for odd cycles

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all $α\in[0,1]$ such that every $n$-vertex $F$-free graph $G$ with minimum degree at least $αn$ has a homomorphic image $H$ of bounded order (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. [Adv. Math. 235 (2013), 261-295]. The homomorphism threshold is less understood and we address the problem for odd cycles.

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arXiv2019

Hypergraphs with vanishing Turán density in uniformly dense hypergraphs

P. Erdős [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of $H$ may depend on the proportion of the considered vertex sets.

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arXiv2019

Loose Hamiltonian cycles forced by large $(k-2)$-degree - sharp version

We prove for all $k\geq 4$ and $1\leq\ell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $\mathcal H$ on $n$ vertices to contain a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$ and $n$ is sufficiently large. This extends a result of Han and Zhao for $3$-uniform hypegraphs.

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arXiv2017

On a Turán problem in weakly quasirandom 3-uniform hypergraphs

Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is weakly $(d,η)$-quasirandom if for any subset $U\subseteq V$ the number of hyperedges of $H$ contained in $U$ is in the interval $d\binom{|U|}{3}\pmη|V|^3$. We show that for any $\varepsilon>0$ there exists $η>0$ such that every sufficiently large weakly $(1/4+\varepsilon,η)$-quasirandom hypergraph contains four vertices spanning at least three hyperedges. This was conjectured by Erdős and Sós and it is known that the density $1/4$ is best possible. Recently, a computer assisted proof of this result based on the flag-algebra method was established by Glebov, Král', and Volec. In contrast to their work our proof presented here is based on the regularity method of hypergraphs and requires no heavy computations. In addition we obtain an ordered version of this result. The method of our proof allows us to study extremal problems of this type in a more systematic way and we discuss a few extensions and open problems here.

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arXiv2017

Monochromatic trees in random graphs

Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs $G$: every $k$-colouring of the edge set of $G$ yields $k$ pairwise vertex disjoint monochromatic trees that partition the whole vertex set of $G$. We determine the threshold for this property for two colours.

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arXiv2017

Some remarks on the extremal function for uniformly two-path dense hypergraphs

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, η,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs $((x,y),(x,z))\in P\times Q$ with $\{x,y,z\}\in E$ is at least $d|\mathcal{K}_{P_2}(P,Q)|-η|V|^3,$ where $\mathcal{K}_{P_2}(P,Q)$ denotes the set of pairs in $P\times Q$ of the form $((x,y),(x,z))$. For a given $3$-uniform hypergraph $F$ we are interested in the infimum $d\geq 0$ such that for sufficiently small $η$ every sufficiently large $(d, η,P_2)$-dense hypergraph $H$ contains a copy of $F$ and this infimum will be denoted by $π_{P_2}(F)$. We present a few results for the case when $F=K_k^{(3)}$ is a complete three uniform hypergraph on $k$ vertices. It will be shown that $π_{P_2}(K_{2^r}^{(3)})\leq \frac{r-2}{r-1}$, which is sharp for $r=2,3,4$, where the lower bound for $r=4$ is based on a result of Chung and Graham [Edge-colored complete graphs with precisely colored subgraphs, Combinatorica 3 (3-4), 315-324].

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arXiv2017

Loose Hamiltonian cycles forced by large $(k-2)$-degree - approximate version

We prove that for all $k\geq 4$ and $1\leq\ell<k/2$, every $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with $δ_{k-2}(\mathcal{H})\geq\left(\frac{4(k-\ell)-1}{4(k-\ell)^2}+o(1)\right)\binom{n}{2}$ contains a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$. This degree condition is asymptotically best possible. The case $k=3$ was addressed earlier by Buß et al.

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arXiv2017

On a generalisation of Mantel's theorem to uniformly dense hypergraphs

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Turán and of Erdős and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$. We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and Sós about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show. Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Král', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises the regularity method for hypergraphs.

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arXiv2016

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

For a given graph $F$ we consider the family of (finite) graphs $G$ with the Ramsey property for $F$, that is the set of such graphs $G$ with the property that every two-colouring of the edges of $G$ yields a monochromatic copy of $F$. For $F$ being a triangle Friedgut, Rödl, Ruciński, and Tetali (2004) established the sharp threshold for the Ramsey property in random graphs. We obtained a simpler proof of this result which extends to a more general class of graphs $F$ including all cycles. The proof is based on Friedgut's criteria (1999) for sharp thresholds, and on the recently developed container method for independent sets in hypergraphs by Saxton and Thomason, and Balogh, Morris and Samotij. The proof builds on some recent work of Friedgut et al. who established a similar result for van der Waerden's theorem.

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arXiv2016

Discrepancy and Eigenvalues of Cayley Graphs

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

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