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Daniela Kühn

Publications and source records attributed to Daniela Kühn.

At least 19 recordsLinked to original sources
arXiv2025

A characterization of testable hypergraph properties

We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability $2/3$ between $k$-graphs that satisfy $P$ and those that are far from satisfying $P$. For the $2$-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the $k$-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the $3$-graph setting. Our proof relies on a random subhypergraph sampling result proved in a companion paper.

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arXiv2024

Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree

In 1977, Erdős asked the following question: for any integers $t,n \in \mathbb{N}$, if $G_1 , \dots , G_n$ are complete graphs such that each $G_i$ has at most $n$ vertices and every pair of them shares at most $t$ vertices, what is the largest possible chromatic number of the union $\bigcup_{i=1}^{n} G_i$? The equivalent dual formulation of this question asks for the largest chromatic index of an $n$-vertex hypergraph with maximum degree at most $n$ and maximum codegree at most $t$. For the case $t = 1$, Erdős, Faber, and Lovász famously conjectured that the answer is $n$, which was recently proved by the authors for all sufficiently large $n$. In this paper, we answer this question of Erdős for $t \geq 2$ in a strong sense, by proving that every $n$-vertex hypergraph with maximum degree at most $(1-o(1))tn$ and maximum codegree at most $t$ has chromatic index at most $tn$ for any $t,n \in \mathbb{N}$. Moreover, equality holds if and only if the hypergraph is a $t$-fold projective plane of order $k$, where $n = k^2 + k + 1$. Thus, for every $t \in \mathbb N$, this bound is best possible for infinitely many integers $n$. This result also holds for the list chromatic index.

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arXiv2024

New bounds on the size of Nearly Perfect Matchings in almost regular hypergraphs

Let $H$ be a $k$-uniform $D$-regular simple hypergraph on $N$ vertices. Based on an analysis of the Rödl nibble, Alon, Kim and Spencer (1997) proved that if $k \ge 3$, then $H$ contains a matching covering all but at most $ND^{-1/(k-1)+o(1)}$ vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all $k > 3$, $H$ contains a matching covering all but at most $ND^{-1/(k-1)-η}$ vertices for some $η= Θ(k^{-3}) > 0$, when $N$ and $D$ are sufficiently large. Our approach consists of showing that the Rödl nibble process not only constructs a large matching but it also produces many well-distributed `augmenting stars' which can then be used to significantly improve the matching constructed by the Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl (1998) and Vu (2000) on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed (2000) on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs).

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arXiv2024

Perfect matchings in random sparsifications of Dirac hypergraphs

For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $δ_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = Ω(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching. Moreover, for every fixed integer $d < k$ and $γ> 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_d(\mathcal H) \geq m_{d}(k,n) + γ\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - Θ(n))$ many perfect matchings, which is best possible up to an $\exp(Θ(n))$ factor.

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arXiv2023

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

A collection of graphs is \textit{nearly disjoint} if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup_{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup_{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy, and we derive this result from a more general list coloring result in the setting of `color degrees' that also implies a result of Reed and Sudakov.

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arXiv2023

Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i, j\in [n]$, then asymptotically almost surely there is an order-$n$ Latin square in which the entry in the $i$th row and $j$th column lies in $L_{i,j}$. The problem of determining the threshold probability for the existence of an order-$n$ Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{ä}ggkvist; our result provides an upper bound which is tight up to a factor of $\log n$ and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a $1$-factorization of a nearly complete regular bipartite graph.

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arXiv2023

A proof of the Erdős-Faber-Lovász conjecture

The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.

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arXiv2022

Hamiltonicity of random subgraphs of the hypercube

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies $δ(H)\geqαn$ with $α>0$ fixed and we consider a random binomial subgraph $\mathcal{Q}^n_p$ of $\mathcal{Q}^n$ with $p\in(0,1]$ fixed, then with high probability $H\cup\mathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. In particular, both results resolve a long standing conjecture, posed e.g. by Bollobás, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. Our techniques also show that, with high probability, for all fixed $p\in(0,1]$ the graph $\mathcal{Q}^n_p$ contains an almost spanning cycle. Our methods involve branching processes, the Rödl nibble, and absorption.

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arXiv2022

Hypergraph regularity and random sampling

Suppose a $k$-uniform hypergraph $H$ that satisfies a certain regularity instance (that is, there is a partition of $H$ given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities). We prove that with high probability a large enough uniform random sample of the vertex set of $H$ also admits the same regularity instance. Here the crucial feature is that the error term measuring the quasirandomness of the subhypergraphs requires only an arbitrarily small additive correction. This has applications to combinatorial property testing. The graph case of the sampling result was proved by Alon, Fischer, Newman and Shapira.

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arXiv2022

Path decompositions of tournaments

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for all sufficiently large tournaments. We also prove an asymptotically optimal result for tournaments of odd order.

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arXiv2022

Almost all optimally coloured complete graphs contain a rainbow Hamilton path

A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_{n}$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_{n}$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.

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arXiv2021

Graph and hypergraph colouring via nibble methods: A survey

This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble' methods. We also give a detailed sketch of some aspects of the recent proof of the Erdős-Faber-Lovász conjecture.

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arXiv2021

Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Hajós states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erdős-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $\frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $\frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $\frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $\max \left\{\frac{odd(G)}{2},\frac{Δ(G)}{2}\right\}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $\frac{Δ(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.

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arXiv2021

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.

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arXiv2020

Counting Hamilton cycles in Dirac hypergraphs

A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. Rödl, Ruciński, and Szemerédi proved that for $k\geq 3$, every $k$-graph on $n$ vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such $k$-graphs is $\exp(n\ln n-Θ(n))$. As a corollary, we obtain a similar estimate on the number of Hamilton $\ell$-cycles in such $k$-graphs for all $\ell\in\{0,\dots,k-1\}$, which makes progress on a question of Ferber, Krivelevich and Sudakov.

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arXiv2020

Dirac's theorem for random regular graphs

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.

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arXiv2020

Euler tours in hypergraphs

We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set.

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