arXiv subjects

Junpeng Zhou

Publications and source records attributed to Junpeng Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Counterexamples to a treewidth conjecture on generalized Tur\'an problems

Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $\chi(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=\Omega(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $\chi(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).

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arXiv2026

Counting large cliques in graphs with a forbidden tree

Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Let $T$ be a tree on $k$ vertices, and write $n=a(k-1)+b$, where $0\leq b<k-1$. Recently, Gerbner and Palmer (Electron. J. Combin., 2026) proposed the following conjecture: for every $r\geq3$, the graph $aK_{k-1}\cup K_b$ maximizes the number of copies of $K_r$ among all $n$-vertex $T$-free graphs. In this paper, we verify their conjecture when $r=k-2$ or $r=k-3\geq5$. More precisely, we show that ${\rm ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r}$ and characterize all extremal graphs.

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arXiv2026

A Spectral Confirmation of the Erd\H{o}s Matching Conjecture

The Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. In this paper, we study a spectral counterpart of this conjecture. For sufficiently large $n$, we determine the maximum spectral radius over all $n$-vertex $r$-uniform hypergraphs whose matching number is less than $s$, and characterize the unique extremal hypergraph. To establish the main theorem, we first apply the shifting method to reduce the problem to shifted hypergraphs. We then derive several spectral upper bounds through hypergraph decomposition and related variational estimates for tensor spectral radii. With these estimates, we analyze the structural properties of shifted-saturated hypergraphs and prove the spectral extremal theorem for shifted hypergraphs with bounded matching numbers. Finally, we drop the shifted condition and extend our spectral bound to general $r$-uniform hypergraphs. Our main theorem states that for any $n$-vertex $r$-uniform hypergraph $H$ with matching number $\nu(H)<s$, the inequality $\rho(H)\leq \rho(\mathcal{F}_{s-1}(n))$ holds whenever $n$ is sufficiently large. Here $\mathcal{F}_{a}(n)$ denotes the family of all $r$-subsets of $[n]$ intersecting the vertex set $[a]$, and equality is attained if and only if $H$ is isomorphic to $\mathcal{F}_{s-1}(n)$. As an immediate corollary, we derive a spectral counterpart of the classical Erd\H{o}s-Ko-Rado theorem for intersecting hypergraph families.

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arXiv2026

On the generalized Tur\'{a}n number of the complete bipartite graph $K_{3,b+1}$

For graphs $F$ and $H$, let $\mathrm{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Very recently, Janzer, Longbrake, and Yepremyan proved that for $3<a\leq b$ and sufficiently large $t$, \begin{equation*} \mathrm{ex}(n,K_{a,b},K_{3,t})=\Theta_{a,b,t}(n^3). \end{equation*} Later, Hou, Hu, and Wang made this threshold explicit by showing that the conclusion holds for all $t\geq 2\max\{3,\lceil b/2\rceil\}+1$. In particular, for every even $b\geq 6$, this matches the necessary threshold $t=b+1$. In this paper, we resolve the remaining case where $b$ is odd. More precisely, we prove that for all fixed integers $b\geq 5$ and $3<a\leq b$, \begin{equation*} \mathrm{ex}(n,K_{a,b},K_{3,b+1})=\Theta_{a,b}(n^3). \end{equation*} Our construction uses a finite-field point set in $\mathrm{PG}(5,q)$ together with an orthogonal polarity. The key new ingredient is the polynomial splitting lemma due to Andrade, Bary-Soroker, and Rudnick, which produces many planes whose intersections with the point set and their polar planes both have size $b$. This gives a $K_{3,b+1}$-free incidence graph while preserving $\Omega_{a,b}(n^3)$ copies of $K_{a,b}$.

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arXiv2026

On the Tur\'an number of blow-ups of $\mathcal{F}_5$

Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Tur\'an number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Tur\'an density. In this paper, we determine the exact Tur\'an number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Tur\'an number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Tur\'an number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Tur\'an hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Tur\'an hypergraph. We also determine the exact Tur\'an number of $\mathcal{F}_{sim}(t)$.

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arXiv2026

Hypergraph extensions of the Alon--Frankl Theorem and rainbow Tur\'an problems

Given a graph $F$, the $r$-expansion $F^{(r)+}$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Recently, Alon and Frankl (JCTB, 2024) and Gerbner (JGT, 2023) studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. Gerbner, Tompkins and Zhou (EJC, 2025) considered the analogous Tur\'{a}n problems on hypergraphs with bounded matching number. In this paper, we study hypergraph extensions of the Alon--Frankl Theorem. More precisely, we determine the maximum number of hyperedges in an $n$-vertex $r$-uniform hypergraph containing neither a matching $M^r_{s+1}$ nor the expansion $K_{\ell+1}^{(r)+}$ of the clique $K_{\ell+1}$ for all small $s<\frac{\ell^2-1}{2}$ and all sufficiently large $s$, respectively. This result partly confirms a conjecture proposed by Gerbner, Tompkins and Zhou (EJC, 2025). As a key tool, we determine the rainbow hyper-Tur\'{a}n number for expansions of cliques, which is defined as the maximum sum of size of a sequence of hypergraphs $\mathcal{H}_1,\dots,\mathcal{H}_k$ that contains no rainbow copies of expansions of cliques with given size. It extends the result of Keevash, Saks, Sudakov and Verstra{\"e}te (AAM, 2004), which determined the rainbow Tur\'an number of cliques in the graph case. These results shows a correlation between the hyper-Tur\'an problem and the rainbow hyper-Tur\'an number.

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arXiv2026

Counting sunflowers in hypergraphs with bounded matching number and Erd\H{o}s Matching Conjecture in the $(t,k)$-norm

It is well known that Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. As a generalization, it is natural to ask for the maximum number of copies of subhypergraphs. Given integers $r\geq2$ and $k\ge 1$, let $S_{r-1,k}^r$ denote the $r$-uniform hypergraph with hyperedges $\{e_1, \dots, e_k\}$ such that there exists an $(r-1)$-set $T$ with $e_i \cap e_j = T$ for $1\le i < j \le k$. We determine the maximum number of copies of $S_{r-1,k}^r$ in an $r$-uniform hypergraph with bounded matching number, and characterize all extremal hypergraphs. An interesting phenomenon is that the extremal numbers and extremal hypergraphs are exactly the same for all $k\ge 1$. Our main tool is the shifting method. By establishing an injection, we prove that the shifting operation does not decrease the number of copies of $S_{r-1,k}^r$ for all $k\geq1$, thereby answering a question raised by Wang and Peng (2026). Moreover, we present a counting method for estimating the number of copies of $S_{r-1,k}^r$ in arbitrary $r$-uniform hypergraphs. Counting the number of copies of $S_{r-1,k}^r$ in $r$-uniform hypergraphs is closely related to Tur\'{a}n problems in the $(r-1,k)$-norm proposed by Chen, Il'kovi\v{c}, Le\'{o}n, Liu and Pikhurko. The $(r-1,k)$-norm of an $r$-uniform hypergraph $\mathcal{H}$ is the sum of the $k$-th power of the degrees $d_{\mathcal{H}}(T)$ over all $(r-1)$-subsets $T \subseteq V(\mathcal{H})$. Combining our established result with that of Frankl (2013), and utilizing the Newton expansion of powers and Stirling numbers of the second kind, we show that Erd\H{o}s Matching Conjecture in the $(r-1,k)$-norm holds, which generalizes the result of Brooks and Linz concerning the $(r-1,2)$-norm case. As a consequence, we obtain a version of the classical Erd\H{o}s--Ko--Rado theorem in the $(r-1,k)$-norm.

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arXiv2026

On the connected Tur\'an number of Berge paths and Berge cycles

Given a graph $F$, a Berge copy of $F$ (Berge-$F$ for short) is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H{o}ri, Salia and Zamora determined the maximum number of hyperedges in a connected $r$-uniform hypergraph on $n$ vertices containing no Berge path of length $k-1$ for all $k\geq 2r+14$ and sufficiently large $n$, and asked for the minimum $k_0$ such that this extremal number holds for all $k\geq k_0$. In this paper, we prove that the extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$, thereby completely resolving the problem posed by Gy\H{o}ri, Salia and Zamora. Moreover, we improve the result of F\"uredi, Kostochka and Luo, who determined the maximum number of hyperedges in a $2$-connected $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$ for all $k\geq 4r$ and sufficiently large $n$, by showing that this extremal number holds for all $k\geq 2r+2$ and fails for $k\le 2r+1$. Our approach reduces Berge-Tur\'an problems to classical extremal graph theory problems, and applies recent work of Ai, Lei, Ning and Shi concerning the feasibility of graph parameters and the Kelmans operation.

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arXiv2026

The Tur\'{a}n number of Berge paths

A Berge path of length $k$ in an $r$-uniform hypergraph is a collection of $k$ hyperedges $h_1,\dots,h_k$ and $k+1$ vertices $v_1,\dots,v_{k+1}$ such that $v_i, v_{i+1}\in h_i$ for each $1\le i\le k$. Gy\H{o}ri, Katona and Lemons [\textit{European J. Combin. 58 (2016) 238--246}] generalized the Erd\H{o}s-Gallai theorem to Berge paths and established bounds for the Tur\'{a}n number of Berge paths. However, these bounds are sharp only when some divisibility conditions hold. Gy\H ori, Lemons, Salia and Zamora [\textit{J. Combin. Theory Ser. B 148 (2021) 239--250}] determined the exact value of the Tur\'{a}n number of Berge paths in the case $k\le r$. In this paper, we settle the final open case $k>r$, thereby completing the determination of the Tur\'{a}n number of Berge paths.

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arXiv2026

On generalized Tur\'{a}n problems for expansions

Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized Tur\'{a}n number, denoted by $\textrm{ex}_r(n,\mathcal{H},\mathcal{F})$, is the maximum number of copies of $\mathcal{H}$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\mathcal{F}$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Tur\'{a}n problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Tur\'{a}n problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Tur\'{a}n number for expansions of complete graphs, and establish the asymptotics of the generalized Tur\'{a}n number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Tur\'{a}n numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.

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arXiv2026

Forbidding edge-critical graphs as trace in uniform hypergraphs

We say a hypergraph $\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = V(G)$ and $\{e \cap S \mid e \in E(\mathcal{H})\}$ contains $G$ as a subgraph. We use $\mathrm{ex}(n, Tr_r(G))$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing $G$ as trace. The study of Tur\'an numbers for traces was initiated by Mubayi and Zhao~(2017) who studied $\mathrm{ex}(n, Tr_r(K_{s+1}))$ where $K_{s+1}$ is a clique on $s+1$ vertices and conjectured the exact value of $\mathrm{ex}(n, Tr_r(K_{s+1}))$. When $r \le s$, the conjecture was covered by a result of Pikhurko~(2013) who gave the exact value of Tur\'an numbers for expanded cliques. Then Gerbner and Picollelli~(2023) gave the exact value for book graphs~($K_{1,1,t}$, the complete tripartite graph with two parts of size one and one part of size $t \ge 2$). We say $G$ is edge-critical if there exists an edge $e \in E(G)$ such that $\chi(G - e) < \chi(G)$ where $\chi(G)$ is the chromatic number of $G$. The definition of edge-critical was given by Simonovits~(1974), who proved that for an edge-critical graph $G$ with $\chi(G) = s+1 \ge 3$, the Tur\'an graph $T(n,s)$ is the unique extremal graph for $ex(n,G)$ as $n$ is sufficiently large. In this paper, we further generalize the results of Gerbner and Picollelli~(2023) to edge-critical graphs. More precisely, we prove that for an edge-critical graph $G$ with $\chi(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Tur\'an graph $T_r(n,s)$ is the unique extremal hypergraph.

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arXiv2025

An efficient proximal algorithm for squared L1 over L2 regularized sparse recovery

In this paper, we consider a squared $L_1/L_2$ regularized model for sparse signal recovery from noisy measurements. We first establish the existence of optimal solutions to the model under mild conditions. Next, we propose a proximal method for solving a general fractional optimization problem which has the squared $L_1/L_2$ regularized model as a special case. We prove that any accumulation point of the solution sequence generated by the proposed method is a critical point of the fractional optimization problem. Under additional KL assumptions on some potential function, we establish the sequential convergence of the proposed method. When this method is specialized to the squared $L_1/L_2$ regularized model, the proximal operator involved in each iteration admits a simple closed form solution that can be computed with very low computational cost. Furthermore, for each of the three concrete models, the solution sequence generated by this specialized algorithm converges to a critical point. Numerical experiments demonstrate the superiority of the proposed algorithm for sparse recovery based on squared $L_1/L_2$ regularization.

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arXiv2025

The Tur\'{a}n number of Berge matchings

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph is $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Let $M_{s+1}$ denote a matching of size $s+1$, i.e., the graph consisting of $s+1$ independent edges. Khormali and Palmer [\textit{European J. Combin.} 102 (2022) 103506] completely determined the Tur\'{a}n number of Berge matchings for sufficiently large $n$. Subsequently, Kang, Ni, and Shan [\textit{Discrete Math.} 345 (2022) 112901] determined the exact value of the Tur\'{a}n number of Berge-$M_{s+1}$ for all $n$ when $r \le s-1$ or $r \ge 2s+2$. In this paper, we settle the final open case $s \le r \le 2s+1$, thereby completing the determination of the Tur\'{a}n number of Berge matchings.

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arXiv2025

On Tur\'{a}n problems for Berge forests

For a graph $F$, an $r$-uniform hypergraph $H$ is a Berge-$F$ if there is a bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph is $\mathcal{F}$-free if it does not contain any member in $\mathcal{F}$ as a subhypergraph. The Tur\'an number of $\mathcal{F}$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-uniform hypergraph on $n$ vertices. In this paper, some exact and general results on the Tur\'{a}n numbers for several types of Berge forests are obtained.

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arXiv2025

On the Tur\'{a}n number of the expansion of the $t$-fan

The $t$-fan is the graph on $2t+1$ vertices consisting of $t$ triangles which intersect at exactly one common vertex. For a given graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by adding $r-2$ distinct new vertices to each edge of $F$. We determine the Tur\'an number of the 3-expansion of the $t$-fan for sufficiently large $n$.

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arXiv2025

On Tur\'{a}n problems for suspension hypergraphs

For a given graph $F$, the $r$-uniform suspension of $F$ is the $r$-uniform hypergraph obtained from $F$ by taking $r-2$ new vertices and adding them to every edge. In this paper, we consider Tur\'{a}n problems on suspension hypergraphs, and we obtain several general and exact results.

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arXiv2025

A min-max reformulation and proximal algorithms for a class of structured nonsmooth fractional optimization problems

In this paper, we consider a class of structured nonsmooth fractional minimization, where the first part of the objective is the ratio of a nonnegative nonsmooth nonconvex function to a nonnegative nonsmooth convex function, while the second part is the difference of a smooth nonconvex function and a nonsmooth convex function. This model problem has many important applications, for example, the scale-invariant sparse signal recovery in signal processing. However, the existing methods for fractional programs are not suitable for solving this problem due to its special structure. We first present a novel nonfractional min-max reformulation for the original fractional program and show the connections between their global (local) optimal solutions and stationary points. Based on the reformulation, we propose an alternating maximization proximal descent algorithm and show its subsequential convergence towards a critical point of the original fractional program under a mild assumption. Moreover, we prove that the proposed algorithm can find an $\epsilon$-critical point of the considered problem within $\mathcal{O}(\epsilon^{-2})$ iterations. By further assuming the Kurdyka-{\L}ojasiewicz (KL) property of an auxiliary function, we also establish the convergence of the entire solution sequence generated by the proposed algorithm. Finally, some numerical experiments on the $L_1/L_2$ least squares problem and scale-invariant sparse signal recovery are conducted to demonstrate the efficiency of the proposed method.

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arXiv2024

On hypergraph Tur\'an problems with bounded matching number

Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. We consider the analogous Tur\'{a}n problems on hypergraphs with bounded matching number, and we obtain some exact results.

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