arXiv subjects

Wenling Zhou

Publications and source records attributed to Wenling Zhou.

16 recordsLinked to original sources

arXiv2026

Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs

A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \emph{$(n,d,\mu)$-dense} if $e_H(X,Y,Z)\ge d|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness. In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,\mu)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $\delta_1(H)$ and the minimum codegree $\delta_2(H)$. We actually prove a general result: define \[ f(d):=\frac{1-\sqrt{(4d-1)/3}}2. \] We prove that $(n,d,\mu)$-density together with $\delta_1(H)\ge\alpha\binom{n-1}{2}$ forces a tight Hamilton cycle whenever $d > 1/3$ and $\alpha>f(d)$. In particular, $f(1/3)=1/3$, which answers Problem~8.3(i) of Ara\'ujo, Piga and Schacht and confirms Conjecture~8.1 of Han, Shu and Wang. For the minimum codegree condition, the sharp diagonal threshold is $(\kappa,\kappa)$, where $\kappa$ is the unique real solution of $\kappa=(1-\kappa)^3$. Since $\kappa\approx0.3177>1/4$, this gives a negative answer to Problem~8.3(ii) of Ara\'ujo, Piga and Schacht and disproves Conjecture~8.2 of Han, Shu and Wang. The two proofs use a common Hamilton-framework reduction, but the two degree conditions lead to distinct dominant-component lemmas for $(n, d, \mu)$-dense $3$-graphs, which are of independent interest and whose proofs do not rely on the absorption method.

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arXiv2026

Pancyclicity of graphs perturbed by a random $F$-factor

We determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor, resolving a conjecture of Espuny D\'iaz and Gir\~ao [Random Structures Algorithms, 2023]. In fact, we prove the stronger pancyclic statement. Let $\alpha^*(K_r)$ and $\alpha_{\text{pan}}^*(K_r)$ denote the Hamiltonicity and pancyclicity thresholds, respectively. We show that $\alpha^*(K_r)=\alpha_{\text{pan}}^*(K_r)=\rho_r$, where $\rho_r$ is the unique positive solution of $x^r+rx-1=0$. The proof is obtained from a general framework for perturbations by a uniformly random $F$-factor, where $F$ is an arbitrary fixed connected graph.

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arXiv2026

Uniform Tur\'an densities of $k$-uniform hypergraphs

For $k\ge 3$, the $(k-2)$-uniform Tur\'an density $\pi_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal F$ of $k$-graphs, we prove that $\pi_{k-2}(\mathcal F)$ equals the corresponding palette Tur\'an density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values \[ \frac{r-1}{r},\quad \frac{(r-1)^2}{r^2},\quad \frac{r-1}{2r},\quad \frac{(k-1)^k}{k^k},\quad \frac{4(k-2)^{k-2}}{k^k},\quad \frac{4(k-2)^{k-2}}{3k^k} \] as $(k-2)$-uniform Tur\'an densities of single $k$-graphs. Finally, for every $k\ge3$, we show that there exist $k$-graphs $F_1,F_2$ such that \[ \pi_{k-2}(\{F_1,F_2\})< \min\{\pi_{k-2}(F_1),\pi_{k-2}(F_2)\}, \] which provides the first examples of \emph{non-principal} families for this density.

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arXiv2026

Extremal problems in uniformly dense hypergraphs and digraphs

The uniform Tur\'an density $\pi_{u}(F)$ of a $3$-uniform hypergraph (or $3$-graph) $F$ is the supremum of all $d$ such that there exist infinitely many $F$-free $3$-graphs $H$ in which every induced subhypergraph on a linearly sized vertex set has edge density at least $d$. Determining $\pi_{u}(F)$ for a given $3$-graph $F$ was proposed by Erd\H{o}s and S\'os in the 1980s, yet only a few cases are known. In particular, it remains open whether $1/2$ can occur as a value of $\pi_{u}$. In this paper, we establish a novel connection between Tur\'an-type extremal problems for digraphs and uniform Tur\'an densities of $3$-graphs. Using digraph extremal results, we give the first verifiable conditions for $3$-graphs $F$ with $\pi_{u}(F) = (r-1)/r$ and $\pi_{u}(F) = (r-1)^2/r^2$ for all $r \ge 2$, and identify the corresponding $3$-graphs. In particular, these $3$-graph classes contain some specific $3$-graphs, such as $K^{(3)-}_4$. We also present a sufficient condition ensuring $\pi_{u}(F)=4/27$ and construct $3$-graphs satisfying it; in particular, our examples are different from the tight $3$-uniform cycles whose uniform Tur\'an density $4/27$ was determined in [{Trans. Amer. Math. Soc. 376 (2023), 4765-4809}]. Finally, we give a short proof of the existence of $3$-graphs $F$ with $\pi_{u}(F)=1/27$, originally established by Garbe, Kr\'al' and Lamaison [{Israel J. Math. 259 (2024), 701-726}] via the hypergraph regularity method.

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arXiv2025

Tur\'an density of stars in uniformly dense hypergraphs

A $3$-uniform hypergraph (or $3$-graph) $H=(V,E)$ is $(d,\mu,1)$-\emph{dense} if for any subsets $X,Y,Z\subseteq V$, the number of triples $(x,y,z)\in X\times Y\times Z$ such that $\{x,y,z\}$ is an edge of $H$ is at least $d|X||Y||Z|-\mu |V|^3$. The \emph{$k$-star} $S_k$ is the $3$-graph with a center vertex and $k$ distinct leaf vertices, whose edge set consists of all triples containing the center and two distinct leaves. Restricting to $dot$-dense $3$-graphs, determining the \emph{$1$-uniform Tur\'an density} $\pi_1(S_k)$ of $S_k$ for $k\ge 4$ was proposed by Schacht in ICM 2022. In particular, Reiher, R\"odl and Schacht gave a palette construction showing that $\pi_1(S_k)\ge \frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 3$, and also proved that $\pi_1(S_3)=1/4$. Lamaison and Wu later showed that this palette construction is optimal for $k\ge 48$. In this paper, we improve the results of Lamaison and Wu by proving that \[ \pi_1(S_k)=\frac{k^2-5k+7}{(k-1)^2} \qquad\text{for all } k\ge 9. \]

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arXiv2025

Degree conditions for spanning expansion hypertrees

The $k$-expansion of a graph $G$ is the $k$-uniform hypergraph obtained from $G$ by adding $k-2$ new vertices to every edge. We determine, for all $k > d \geq 1$, asymptotically optimal $d$-degree conditions that ensure the existence of all spanning $k$-expansions of bounded-degree trees, in terms of the corresponding conditions for loose Hamilton cycles. This refutes a conjecture by Pehova and Petrova, who conjectured that a lower threshold should have sufficed. The reason why the answer is off from the conjectured value is an unexpected `parity obstruction': all spanning $k$-expansions of trees with only odd degree vertices require larger degree conditions to embed. We also show that if the tree has at least one even-degree vertex, the codegree conditions for embedding its $k$-expansion become substantially smaller.

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arXiv2025

Hypergraphs with a quarter uniform Turán density

The uniform Turán density $π_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $π_{1}(F)$ for given hypergraphs $F$ was suggested by Erdős and Sós in 1980s. In particular, they raised the questions of determining $π_{1}(K_4^{(3)-})$ and $π_{1}(K_4^{(3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{(3)-}$, there are very few hypergraphs whose uniform Turán density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $π_{1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Turán density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F^{\star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Turán density is also $1/4$.

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arXiv2025

A strengthening on consecutive odd cycles in graphs of given minimum degree

Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when $k$ is even. In this paper, we confirm this conjecture for any $k\in \mathbb N$. Moreover, we also improve some previous results about cycles of consecutive lengths.

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arXiv2023

The minimum positive uniform Turán density in uniformly dense $k$-uniform hypergraphs

A $k$-graph (or $k$-uniform hypergraph) $H$ is uniformly dense if the edge distribution of $H$ is uniformly dense with respect to every large collection of $k$-vertex cliques induced by sets of $(k-2)$-tuples. Reiher, Rödl and Schacht [Int. Math. Res. Not., 2018] proposed the study of the uniform Turán density $π_{k-2}(F)$ for given $k$-graphs $F$ in uniformly dense $k$-graphs. Meanwhile, they [J. London Math. Soc., 2018] characterized $k$-graphs $F$ satisfying $π_{k-2}(F)=0$ and showed that $π_{k-2}(\cdot)$ ``jumps" from 0 to at least $k^{-k}$. In particular, they asked whether there exist $3$-graphs $F$ with $π_{1}(F)$ equal or arbitrarily close to $1/27$. Recently, Garbe, Král' and Lamaison [arXiv:2105.09883] constructed some $3$-graphs with $π_{1}(F)=1/27$. In this paper, for any $k$-graph $F$, we give a lower bound of $π_{k-2}(F)$ based on a probabilistic framework, and provide a general theorem that reduces proving an upper bound on $π_{k-2}(F)$ to embedding $F$ in reduced $k$-graphs of the same density using the regularity method for $k$-graphs. By using this result and Ramsey theorem for multicolored hypergraphs, we extend the results of Garbe, Král' and Lamaison to $k\ge 3$. In other words, we give a sufficient condition for $k$-graphs $F$ satisfying $π_{k-2}(F)=k^{-k}$. Additionally, we also construct an infinite family of $k$-graphs with $π_{k-2}(F)=k^{-k}$.

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arXiv2023

Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

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arXiv2022

Tiling multipartite hypergraphs in Quasi-random Hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $Ω(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $Ω(n^2)$ and minimum codegree $Ω(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense quasi-random $3$-graphs of order $n$ with minimum codegree $Ω(n)$ and $f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs given a minimum codegree condition $Ω(n)$. Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree $Ω(n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in quasi-random $3$-graphs given a minimum codegree condition $Ω(n)$.

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arXiv2022

Integer colorings with no rainbow $k$-term arithmetic progression

In this paper, we study the rainbow Erdős-Rothschild problem with respect to $k$-term arithmetic progressions. For a set of positive integers $S \subseteq [n]$, an $r$-coloring of $S$ is \emph{rainbow $k$-AP-free} if it contains no rainbow $k$-term arithmetic progression. Let $g_{r,k}(S)$ denote the number of rainbow $k$-AP-free $r$-colorings of $S$. For sufficiently large $n$ and fixed integers $r\ge k\ge 3$, we show that $g_{r,k}(S)<g_{r,k}([n])$ for any proper subset $S\subset [n]$. Further, we prove that $\lim_{n\to \infty}g_{r,k}([n])/(k-1)^n= \binom{r}{k-1}$. Our result is asymptotically best possible and implies that, almost all rainbow $k$-AP-free $r$-colorings of $[n]$ use only $k-1$ colors.

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arXiv2022

$F$-factors in Quasi-random Hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the $F$-factor problem in quasi-random $k$-graphs with minimum degree $Ω(n^{k-1})$. They posed the problem of characterizing the $k$-graphs $F$ such that every sufficiently large quasi-random $k$-graph with constant edge density and minimum degree $Ω(n^{k-1})$ contains an $F$-factor, and in particular, they showed that all linear $k$-graphs satisfy this property. In this paper we prove a general theorem on $F$-factors which reduces the $F$-factor problem of Lenz and Mubayi to a natural sub-problem, that is, the $F$-cover problem. By using this result, we answer the question of Lenz and Mubayi for those $F$ which are $k$-partite $k$-graphs, and for all 3-graphs $F$, separately. Our characterization result on 3-graphs is motivated by the recent work of Reiher, Rödl and Schacht [J. Lond. Math. Soc., 2018] that classifies the 3-graphs with vanishing Turán density in quasi-random $k$-graphs.

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arXiv2021

Large rainbow matchings in edge-colored graphs with given average color degree

A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of different colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(2020)] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that |V(G)|>=12k^2+4k, or G is a properly edge-colored graph with |V(G)|>=8k. In this paper, we show that every edge-colored graph G with |V(G)|>=4k-4 and average color degree at least 2k-1 contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least 2k-1 contains a rainbow matching of size at least k. The bound is sharp for complete graphs.

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arXiv2021

The Turan problems of directed paths and cycles in digraphs

Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orientation of $C_4$. For all other orientations of $C_4$, we also study the maximum size and the extremal digraphs avoiding them.

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arXiv2020

Wide-Sense 2-Frameproof Codes

Various kinds of fingerprinting codes and their related combinatorial structures are extensively studied for protecting copyrighted materials. This paper concentrates on one specialised fingerprinting code named wide-sense frameproof codes in order to prevent innocent users from being framed. Let $Q$ be a finite alphabet of size $q$. Given a $t$-subset $X=\{x ^1,\ldots, x ^t\}\subseteq Q^n$, a position $i$ is called undetectable for $X$ if the values of the words of $X$ match in their $i$th position: $x_i^1=\cdots=x_i^t$. The wide-sense descendant set of $X$ is defined by $\wdesc(X)=\{y\in Q^n:y_i=x_i^1,i\in {U}(X)\},$ where ${U}(X)$ is the set of undetectable positions for $X$. A code ${\cal C}\subseteq Q^n$ is called a wide-sense $t$-frameproof code if $\wdesc(X) \cap{\cal C} = X$ for all $X \subseteq {\cal C}$ with $|X| \le t$. The paper improves the upper bounds on the sizes of wide-sense $2$-frameproof codes by applying techniques on non $2$-covering Sperner families and intersecting families in extremal set theory.

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