arXiv subjects

Jianfeng Hou

Publications and source records attributed to Jianfeng Hou.

At least 19 recordsLinked to original sources

arXiv2026

The exact digirth and out-degree bounds for $C(k,k)$-subdivision and out-forest in digraphs

For each integer $k\ge2$, let $C(k,k)$ denote the oriented cycle formed by two internally vertex-disjoint directed paths of length $k$ with the same initial and terminal vertices. Picasarri-Arrieta and Rambaud ({\small European J. Combin., 2024}) asked for the smallest integer $g_k$ such that every digraph with minimum out-degree at least two and digirth at least $g_k$ contains a subdivision of $C(k,k)$. We completely resolve this problem by proving that $g_k=k+1$. Our proof relies on a refinement of Thomassen's theorem ({\small Discrete Math., 1985}) on two directed cycles intersecting in exactly one vertex: we prove that every digraph with minimum out-degree at least two contains two directed cycles intersecting in exactly one arc. We also establish an embedding theorem for out-forests with prescribed images of their roots, under minimum out-degree and digirth conditions determined by the number of leaves and the heights of vertices in antichains. As an application, for integers $k,\ell\ge2$, we show that every digraph with minimum out-degree at least $k$ and digirth at least $k(\ell-1)+1$ contains $k$ directed paths of length $\ell$ sharing only their initial vertex, which can be prescribed arbitrarily. This improves the previous digirth bound $(k^\ell-1)/(k-1)+1$ of Picasarri-Arrieta and Rambaud to a sharp bound.

math.CO↗

arXiv2026

Density Hajnal--Szemerédi theorem for cliques of size four

The exact minimum-degree thresholds that guarantee the existence of $k$ vertex-disjoint copies of the complete graph $K_r$ in an $n$-vertex graph were determined in the celebrated theorems of Dirac (1952) for $r=2$, Corrádi--Hajnal (1963) for $r=3$, and Hajnal--Szemerédi (1970) for general $r \ge 4$. A natural and fundamental problem is to determine the corresponding average-degree (or density) thresholds. This problem turns out to be substantially more difficult. The seminal theorem of Erdős--Gallai (1959) resolved the case $r=2$. The case $r=3$ was settled for large $n$ by Allen, Böttcher, Hladký, and Piguet~\cite{ABHP15}, who remarkably identified four distinct classes of extremal constructions arising in different ranges of $k$. They further posed the problem of establishing a density analogue of the Hajnal--Szemerédi theorem, noting that ``We are not even sure what the complete family of extremal graphs should be.'' In this paper, we establish an asymptotic density Hajnal-SzemerHedi theorem for $K_4$ by determining $\mathrm{ex}(n,(k+1)K_4)$ up to an additive $O(n)$ term for all integers $n\ge 4k\ge 0$. Five families of constructions show that this bound is sharp up to an additive $O(n)$ term. For general $r\ge 5$, we propose a candidate family consisting of $r+1$ extremal construction classes.

math.CO↗

arXiv2026

On decomposition thresholds for odd cycles

We determine the asymptotic minimum-degree threshold for decompositions into cycles of a fixed odd length. We prove that, for each odd integer $\ell\ge3$, every $\eps>0$ and sufficiently large $n$, every $n$-vertex $C_\ell$-divisible graph $G$ with minimum degree $δ(G)\ge (\ell/(2\ell-2)+\eps)n$ has a $C_\ell$-decomposition, where $C_{\ell}$ denotes the cycle of length $\ell$. This settles the remaining cases $\ell=5,7,\ldots,71$ and completes the determination of the thresholds for all odd cycles. Our main ingredient is a sharp fractional decomposition theorem for weighted triangles: for every real $a\ge 1$,every $n$-vertex graph $G$ with $δ(G)\ge \left(\frac12+\frac{1}{2(a+1)}\right)n$ admits a fractional decomposition into triangles with edge weights $a,1,1$.

math.CO↗

arXiv2026

A near-quadratic lower bound for sets with no unique sums

Let $m(p)$ be the least size of a subset of $\F_p$ with at least two elements for which every sum has two distinct representations as unordered pairs, allowing repetition. We prove that, for every prime $p\ge64$, \[ m(p)\ge2^{-80}\left(\frac{\log p}{\log\log p}\right)^2. \] The argument compresses the full integer collision lattice by unit-pivot elimination. A shared random sample and forests of bounded diameter give $O(\sqrt n+n/\log p)$ surviving coordinates of polynomial height for a minimal set of size $n$. A nonzero minor divisible by $p$ then gives the lower bound. We also construct weakly ternary-balanced seeds yielding \[ m(p)\le\frac{(\log p)^2}{2(\log3)^2} +\left(\frac1{4\log3}+o(1)\right) \frac{(\log p)^2}{\log\log p}. \] Consequently $m(p)=(\log p)^{2+o(1)}$ as $p$ tends to infinity through the primes. The constant-factor order of $m(p)$ remains undetermined.

math.CO↗

arXiv2026

Vertex-colored Turán theorems with applications in extremal hypergraph problems

Balogh, Clemen, and Lidický proved that the $\ell_{2}$-norm Turán problem for $K_{5}^{3}$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_{5}^{3}$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an $\ell_{2}$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_{4}$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.

math.CO↗

arXiv2026

Exact Turán number of the Fano plane in the $\ell_2$-norm

A classical object in hypergraph Turán theory is the Fano plane $\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Turán density and exact Turán number of $\mathbb{F}$, first proposed as a problem by Sós~\cite{Sos76} in the 1970s, were determined through a sequence of works by De Caen--Füredi~\cite{DCF00}, Füredi--Simonovits~\cite{FS05}, Keevash--Sudakov~\cite{KS05}, and Bellmann--Reiher~\cite{BR19}. Confirming a conjecture of Balogh--Clemen--Lidický~{\cite[Conjecture~3.1]{BCL22a}} in a stronger form, we establish an Andrásfai--Erdős--Sós--type stability theorem for $\mathbb{F}$ in the $\ell_2$-norm: there exists a positive constant $\varepsilon$ such that for large $n$, every $\mathbb{F}$-free $3$-graph on $n$ vertices with minimum $\ell_2$-norm degree at least $(5/4 - \varepsilon)n^3$ must be bipartite. As a consequence, for large $n$, the balanced complete bipartite $3$-graph is the unique extremal construction for the $\ell_{2}$-norm Turán problem of $\mathbb{F}$. Our proof includes a refinement of a classical result by Ahlswede--Katona~\cite{AK78} on counting stars, and the establishment of an Andrásfai--Erdős--Sós--type theorem for a multigraph Turán problem studied by Bellmann--Reiher~\cite{BR19}, both of which are of independent interest.

math.CO↗

arXiv2026

Simultaneous Residue-Class Selection in Prescribed-Difference Packings

A family $\mathcal F$ of finite subsets of $\mathbb Z$ is packed into $[N]$ if suitable integer translates of its members are pairwise disjoint subsets of $[N]$. We study two prescribed-difference packing problems of Alon, Dębski, Grytczuk and Przybyło for the arithmetic progressions $A_d=\{d,2d,\ldots,\lfloor n/d\rfloor d\}$ and $B_d=\{d,2d,\ldots,nd\}$. Let $m(n)$ and $M(n)$ denote the corresponding minimum packing lengths, with subscripts indicating restrictions on the admissible differences, and let $\mathcal P(x)$ denote the set of primes at most $x$. The key ingredient is a residue-class selection scheme that encodes pairwise intersection constraints by cyclic intervals. A lattice-covering argument yields a simultaneous admissible choice, permitting substantial overlap of the containing intervals and providing the sharp upper bounds needed for the bounded-diameter family and the prime-difference equal-cardinality family. For the bounded-diameter family, we prove $m(n)\sim m_{\mathcal P(\sqrt n)}(n)\sim 4n^{3/2}/(3\log n)$. For the equal-cardinality family with prime differences, we prove $M_{\mathcal P(n)}(n)\sim n^3/(6\log n)$. For the full equal-cardinality family, we show $M(n)\ge \left(\frac{19}{108}-o(1)\right)n^3/\log n$. Together with corresponding estimates for restricted ranges of differences, these results prove several conjectures of Alon--Dębski--Grytczuk--Przybyło and disprove others.

math.NT↗

arXiv2026

Equal subset sums and close divisors

For $k\geq2$, let $α_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $β_k$ be the supremum of the $c<1$ for which, with probability tending to one as $D\to\infty$, the set $\mathbf A\cap[D^c,D]$ has $k$ distinct subsets with the same sum. We prove that $α_k=β_k/(1-β_k)$, resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027--1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.

math.NT↗

arXiv2026

Local versus global average degree in bipartite graphs

Let $F_{\mathrm{bip}}(n)$ denote the maximum, over all $n$-vertex bipartite graphs without isolated vertices, of the ratio of the minimum local average degree to the global average degree. We prove that $F_{\mathrm{bip}}(n)=\frac14\sqrt n+\frac38+o(1)$. This answers a problem posed by Tuza. The lower bound is obtained from a new family of bipartite constructions, while the upper bound is proved by a new algebraic approach, different from the combinatorial methods used previously. The key ingredient is a spectral argument based on a normalized biadjacency matrix.

math.CO↗

arXiv2026

The Erdős Matching Conjecture for 4-uniform hypergraphs

We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number $s\ge6004$. Specifically, for integers $s\ge 6004$ and $n\ge 4s+4$, if a $n$-vertex 4-uniform hypergraph $\mathcal F$ does not contain a matching of size $s+1$, then \[ |\mathcal F|\le \max\left\{\binom{4s+3}{4},\binom n4-\binom{n-s}{4}\right\}. \] Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.

math.CO↗

arXiv2026

Feedback edge set in bipartite digraph

Let $β(G)$ denote the minimum size of a feedback edge set of a digraph $G$, and let $γ(G)$ denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky--Seymour--Sullivan conjecture for $3$-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, $γ(G)$ is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every $4$-free bipartite digraph $G$ satisfies $β(G)\le γ(G)/2$. We also determine the exact Turán number of $2k$-free strong bipartite digraphs with partite sets $X$ and $Y$: if $|X|,|Y|\ge k+1$, then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case $k=2$, we analyze the structure of $4$-free strong bipartite Turán digraphs and prove the sharper bound $β(G)\le γ(G)/3$ for all such digraphs. This constant is attained by a natural balanced three-block construction.

math.CO↗

arXiv2026

An Improved Upper Bound for Finite Sidon Sets via Vector-Valued Smoothing

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0,1,\ldots,N-1\}$. We prove \[ F(N)\le N^{1/2}+γ_0N^{1/4}+O(1), \qquad γ_0=0.94349\ldots<0.9435. \] This improves the previously published coefficient $0.98183$. Our argument develops a vector-valued smoothing method that combines several discrete smoothing kernels, each accompanied by boundary weights that compensate for endpoint effects, so that their weighted combination satisfies the required finite covering inequalities. We also show that averaging systems that satisfy these inequalities individually cannot improve upon the best constituent. Numerical optimization is used to find an eight-component candidate system, which is then certified by exact rational arithmetic.

math.CO↗

arXiv2026

On the Erdős Five-Edge Intersection Problem

For an $n$-vertex graph $G$ and a permutation $π$ of its vertex set, let \[ I_G(π)=|E(G)\cap E(G_π)|,\qquad μ(G)=\min_π I_G(π), \] where $G_π$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $π(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $μ(G)\ge k$. Erdős recorded a construction of Mullin showing $f(n,5)\le 2n-2$ and asked whether equality holds for sufficiently large $n$. We prove that it does: \[ f(n,5)=2n-2 \] for all sufficiently large $n$. The proof strategy is a core--buffer--completion framework: it moves the few high-degree vertices into carefully chosen low-degree positions, confines the allowed overlap to this bounded part, and then relabels the sparse remainder without creating any additional common edge.

math.CO↗

arXiv2026

On the Generalized Rational Exponents Conjecture

For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $α\ge1$, there exist fixed graphs $H_α$ and $F_α$ such that \[ \ex(n,H_α,F_α)=Θ(n^α). \] Furthermore, the counting graph $H_α$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization--compression--shift framework, which transforms the Bukh--Conlon finite family construction for edges into a generalized Turán problem setting with a single forbidden graph.

math.CO↗

arXiv2026

Non-trivial Intersection Problems for Multi-partite Hypergraphs

We study non-trivial intersection problems for multi-partite hypergraphs, excluding the usual extremal examples determined by fixed vertices or fixed coordinates. Our first result determines the exact value of the non-trivial $t$-intersection problem in the symmetric product $[n]^r$ for $1\le t\le r-2$ and all $n\ge2$. Frankl and Nie proved a two-candidate formula for sufficiently large $n$ and conjectured it for all $n\ge 2$; our formula shows that the conjectured expression must be enlarged, in small ranges of $n$, by additional ball-type terms arising from the Frankl families. Our second result concerns intersecting families in general products $X_1\times\cdots\times X_r$, where $|X_i|=n_i$, with no common vertex. Let $m_0(1,n_1,\ldots,n_r)$ denote the largest size of such a family. We show that this number is equal to the maximum of $\sum_{X\in \mathcal{D}}\prod_{i\in X}(n_i-1)$ over all downsets $\mathcal{D}\subseteq 2^{[r]}$ such that $\bigcup_{X\in \mathcal{D}}X=[r]$ and no two members of $\mathcal{D}$ have union $[r]$. This finite reduction separates the intersection obstruction from the part sizes and yields explicit fully asymmetric formulas for $r=4,5,6$.

math.CO↗

arXiv2026

Revisiting Degree-Corrected Spectral Clustering: a Condition-Free Spectral Analysis and Extension

Spectral clustering is a representative graph clustering technique with strong interpretability and theoretical guarantees. Degree-corrected spectral clustering (DCSC) has emerged as the state-of-the-art for this technique. While prior studies have provided impressive theoretical insights for DCSC, their analyses typically depend on specific probabilistic frameworks (e.g., stochastic block models) and conditions. In this study, we explore an alternative condition-free analysis for the clustering quality of DCSC from a pure spectral view, without any random graph models. It gives bounds for the number of mis-clustered nodes w.r.t. the optimal partition of conductance minimization while involving quantities that indicate impacts of (\romannumeral1) degree heterogeneity and (\romannumeral2) weakness of clustering structures to the clustering quality. Inspired by graph neural networks (GNNs) and their over-smoothing effect, we propose ASCENT (Adaptive Spectral ClustEring with Node-wise correcTion), a simple yet effective extension of DCSC. Different from most DCSC methods with a constant degree correction, ASCENT follows a node-wise correction scheme. It can assign different corrections for nodes via a GNN mean aggregator. We demonstrate that (\romannumeral1) ASCENT reduces to conventional DCSC methods when encountering over-smoothing; (\romannumeral2) some early stages before over-smoothing can potentially result in better clustering quality.

cs.SI↗

arXiv2026

Turán numbers of $4$-uniform tight even cycles minus one edge

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Turán number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash--Sudakov on the Turán density and the Turán number of the $4$-uniform expanded triangle. We also show that the Turán density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Turán density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.

math.CO↗

arXiv2026

Fixed-density profiles for the semi-induced 4-vertex star

We study the fixed-density semi-inducibility profiles of the red-blue star $S_{2,1}$, which has one distinguished center, two red edges and one blue edge. For an $n$-vertex graph $G$, let $N(S_{2,1},G)$ be the number of injective labeled copies in which the two red edges of $S_{2,1}$ are mapped to edges of $G$ and its blue edge is mapped to a non-edge of $G$, that is, \begin{align*} N(S_{2,1},G)= \sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)). \end{align*} For every fixed red edge density $β\in[0,1]$, we determine both extremal $S_{2,1}$-densities. On the upper side, we prove the missing low-density range and, together with the theorem of Balogh, Lidický, Mubayi, Pfender and Volec for $β\ge 1/4$, obtain the full four-branch profile predicted in their work. On the lower side, we show that the natural endpoint profile coming from the quasi-star and quasi-clique constructions is not universal; the correct minimum is given by a one-parameter three-class complement-split family. The proofs use a transfer argument with degree-square tie-breaking, reducing the extremal analysis to almost-regular, threshold and finite-staircase optimizations.

math.CO↗