arXiv subjects

Jianfeng Hou

Publications and source records attributed to Jianfeng Hou.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotically optimal packings of arithmetic progressions with prime differences

For a positive integer $n$, put $A_d=\{id:1\le i\le\lfloor n/d\rfloor\}$ for $1\le d\le n$ and $B_d=\{id:1\le i\le n\}$ for $d\in\mathbb{N}$. For $D\subseteq\{1,\ldots,n\}$, let $m_D(n)$ be the minimum length of an integer interval containing pairwise disjoint shifted copies of $A_d$ for all $d\in D$. For a finite set $E\subseteq\mathbb{N}$, define $M_E(n)$ analogously using $B_e$, $e\in E$. Let $\mathcal{P}(x)=\{p\le x:p\text{ is prime}\}$. We prove $m_{\mathcal{P}(\sqrt n)}(n)=\left(\frac43+o(1)\right)\frac{n^{3/2}}{\ln n}$ and $M_{\mathcal{P}(n)}(n)=\left(\frac16+o(1)\right)\frac{n^3}{\ln n}$ as $n\to\infty$. These asymptotic formulas attain the known lower bounds and settle two conjectures of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o concerning prime differences. The proof combines a cyclic phase-selection principle with lattice covering estimates and a decomposition into regular blocks of primes.

math.NT↗

arXiv2026

Feedback edge set in bipartite digraph

Let $\beta(G)$ denote the minimum size of a feedback edge set of a digraph $G$, and let $\gamma(G)$ 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, $\gamma(G)$ is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every $4$-free bipartite digraph $G$ satisfies $\beta(G)\le \gamma(G)/2$. We also determine the exact Tur\'an 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 Tur\'an digraphs and prove the sharper bound $\beta(G)\le \gamma(G)/3$ for all such digraphs. This constant is attained by a natural balanced three-block construction.

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 $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\alpha$ such that \[ \ex(n,H_\alpha,F_\alpha)=\Theta(n^\alpha). \] Furthermore, the counting graph $H_\alpha$ 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 Tur\'an problem setting with a single forbidden graph.

math.CO↗

arXiv2026

On the Erd\H{o}s Five-Edge Intersection Problem

For an $n$-vertex graph $G$ and a permutation $\pi$ of its vertex set, let \[ I_G(\pi)=|E(G)\cap E(G_{\pi})|,\qquad \mu(G)=\min_{\pi} I_G(\pi), \] where $G_{\pi}$ is the copy of $G$ obtained by relabelling every vertex $x\in V(G)$ as $\pi(x)$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph $G$ satisfying $\mu(G)\ge k$. Erd\H{o}s 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

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

Local and 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.

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 \[ F(N)\le N^{1/2}+\gamma_0N^{1/4}+O(1), \qquad \gamma_0=0.94349\ldots<0.9435. \] 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

Tur\'an 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 Tur\'{a}n 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 Tur\'an density and the Tur\'{a}n number of the $4$-uniform expanded triangle. We also show that the Tur\'{a}n 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 Tur\'{a}n 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 $\beta\in[0,1]$, 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, Lidick\'y, Mubayi, Pfender and Volec for $\beta\ge 1/4$, 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↗

arXiv2026

A Spectral Theory of Normalized Corrected GNN Propagation

We develop a spectral theory for \emph{normalized corrected GNN propagation}. The object of study is the symmetric normalized adjacency with its degree-stationary component removed, matching the normalization used by standard GCN-style models while isolating the stationary direction most directly tied to oversmoothing. The central theoretical question is whether this corrected normalized operator preserves class-discriminative signal after many propagation layers. Our main result is a high-probability exact-recovery theorem for the binary Contextual Stochastic Block Model after $k=O(\log n)$ propagation steps in the dense polylogarithmic regime $p\ge C\log^B n/n$, for any fixed $B>4$, under explicit graph-signal and feature-SNR conditions. We also establish a multi-class partial recovery theorem showing contraction toward class centers for most nodes. Synthetic and real node-classification experiments are included as empirical checks of the theory's predicted dependence on depth, graph signal, and feature noise.

cs.LG↗

arXiv2026

Explicit thresholds in a generalized Tur\'an problem for $K_{3,t}$-free graphs

For graphs $F$ and $H$, let $\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Janzer, Longbrake and Yepremyan recently proved that, for fixed $3<a\le b$ and sufficiently large $t$, \[ \ex(n,K_{a,b},K_{3,t})=\Theta(n^3). \] We make their threshold explicit, showing that this conclusion holds for all $t\ge \tau(b):=2\max\{3,\lceil b/2\rceil\}+1.$ In particular, for every even $b\ge 6$, this matches the necessary threshold $t=b+1$. The main new ingredient is an explicit finite-field point set whose plane sections are controlled directly, rather than through a general bounded-complexity algebraic lemma. This direct line-and-conic section analysis gives the required $K_{3,t}$-freeness while preserving many coplanar $b$-element subsets.

math.CO↗

arXiv2026

On saturation problems involving clique number and matching number

For a clique $K_r$, a graph is $K_r$-saturated if it contains no copy of $K_r$ and the addition of any edge from its complement creates a $K_r$. A classical result of Erd\H{o}s-Hajnal-Moon and Zykov shows that the number of edges of an $n$-vertex $K_r$-saturated graph is at least $(r-2)n-\binom{r-1}{2}$. In this paper, we focus on the number of edges of the $K_r$-saturated graphs with a fixed matching number. Let $G$ be an $n$-vertex $K_r$-saturated graph with matching number $\nu(G) = s$. For sufficiently large $n$, we prove that the number of edges \begin{equation*} e(G)\geq \left\{\begin{array}{cl}{(r-1)n-\frac{r}{2}(r-1)-1,}&{\quad\mathrm{if}~s=r-1;}\\{(r-1)n + (s-r)^2 - \frac{1}{2}(r+2)(r-3) - 5,}&{\quad\mathrm{if}~s>r-1.}\\\end{array}\right. \end{equation*} Moreover, we completely characterize the graphs attaining the equality.

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

On cliques in hypergraphs under bounded $(j,p)$-norm

Let $\mathcal{H}$ be an $r$-uniform hypergraph. For $S\in \binom{V(\mathcal{H})}{j}$, let $\mathrm{deg}(S)$ be the number of edges of $\mathcal{H}$ containing $S$, and define the $(j,p)$-norm of $\mathcal{H}$ by $\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\mathrm{deg}(S)^p\right)^{1/p}$. Motivated by a problem of Chao, Dong, Shen and Yang, we determine the maximum number of $t$-cliques in an $n$-vertex $r$-graph with bounded $(j,p)$-norm in the range $p>(t-j)/(r-j)$. The proof uses an entropy argument adapted to hypergraphs, together with a continuous interpolation step. The bound is sharp whenever the corresponding Steiner systems exist.

math.CO↗

arXiv2026

Vertex-colored Tur\'{a}n theorems with applications in extremal hypergraph problems

Balogh, Clemen, and Lidick\'{y} proved that the $\ell_{2}$-norm Tur\'{a}n 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 Tur\'{a}n-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

A finite-board reduction for the Erd\H{o}s Matching Conjecture and the 4-uniform case via exact certificates

We prove the 4-uniform Erd\H{o}s Matching Conjecture for every matching number $s\ge 6961$. The proof has two parts. First, building on ideas from Frankl--R\"odl--Ruci\'nski, we formulate a general finite-board criterion for the $r$-uniform conjecture. The criterion has two assumptions: the $(r-1)$-uniform cover-side bound for links with matching number at most $t$ holds at every $m\ge n_r(t)$, and a finite optimization problem for mixed-size trace configurations on an $(r^2+r-1)$-vertex board. Together with the corresponding lower-uniformity input, this finite-board optimization implies the Erd\H{o}s Matching Conjecture with explicit large-matching thresholds. Second, we verify the finite-board assumption for $r=4$. The local board has 19 vertices, and the required inequality is decomposed into three weighted local inequalities: a leading wide layer, a 15-board layer, and an 11-board layer. The verification is reduced to exact finite optimization and certificate-validation problems: Ferrers down-set enumerations for pair and triple traces, rational Farkas-dual certificates for the top-star branch, integer branch-and-bound up-set hitting and pattern searches for the no-top-star branch, and residual-cut dual certificates for the 15-board and 11-board layers.

math.CO↗

arXiv2026

Near-Optimal Algorithms for Maximal Clique Enumeration in Structurally Sparse Graphs

We study the exact enumeration of maximal cliques in graph classes defined by excluded clique minors and excluded clique immersions. For n-vertex K_t-minor-free graphs, we give an algorithm that lists all maximal cliques in n * 4^(2t/5+o(t)) time, significantly improving the previous n * 2^O(t log log t) bound of Eppstein, L\"offler, and Strash. For n-vertex K_t-immersion-free graphs, we establish the first exact enumeration algorithm parameterized by immersion number, achieving a running time of n * 3^(t/3+o(t)). While both algorithms employ a common degeneracy-based root-assignment scheme, their analyses require distinct structural mechanisms. Crucially, rather than applying generic sparsity bounds, our algorithms deeply integrate the specific structural obstructions -- local density thresholds for minors and minimum-degree branchings for immersions -- directly into the enumeration logic. We also prove matching output-size lower bounds, up to sub-exponential factors in t, using specialized constructions. Consequently, the exponential bases 4^(2/5) and 3^(1/3) are asymptotically optimal.

cs.DS↗

arXiv2026

Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs

Let $ t\ge s\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $Ω(d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erdős asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree $d$ is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least $(\frac{1}{2} -o_d(1)) \log d$. In this paper, we improve this low bound to $\left(\frac{s}{2(s-1)} -o_d(1)\right) \log d$ for $K_{s, t}$-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.

math.CO↗