arXiv subjects

Mei Lu

Publications and source records attributed to Mei Lu.

At least 19 recordsLinked to original sources

arXiv2026

DPSF-Net: A Dual-Prior Spatial-Frequency Network for Real-World Remote Sensing Image Dehazing

Real-world remote sensing image dehazing (RSID) remains challenging because atmospheric scattering, spatially non-uniform haze and colour distortion jointly degrade structural and spectral information. Most deep learning methods rely on RGB inputs and spatial-domain feature extraction, which limits their ability to separate global background haze from local surface details. Here, we propose DPSF-Net, a dual-prior spatial-frequency network built on MCAF-Net for real-world RSID. The network uses hazy RGB images and dark channel prior (DCP) maps as joint inputs, allowing physical degradation cues to guide end-to-end feature learning. A spatial-frequency residual interaction block introduces a FourierUnit branch into multi-directional spatial interaction to model large-scale haze components. A prior-guided feature attention module adaptively fuses prior and attention features to reduce colour shift and structural distortion. A selective kernel complementary fusion module screens multi-scale skip features through bidirectional residual complementary gating and selective kernel fusion. Extensive experiments demonstrate that DPSF-Net achieves state-of-the-art performance on the real-world RRSHID remote sensing image dehazing benchmark and remains competitive across multiple synthetic datasets. Moreover, the proposed method strikes a favourable balance among restoration quality, parameter count and computational complexity, supporting the effectiveness of dual-prior spatial-frequency modelling.

cs.CV↗

arXiv2026

A Sharp Spectral Erd\H{o}s--Ko--Rado Theorem for Uniform Hypergraphs

The spectral Erd\H{o}s--Ko--Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $\rho_r$ for their spectral radii. For $2\le t 2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $\rho_0\ge\rho_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede--Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.

math.CO↗

arXiv2026

Entropy Transference for Rainbow-$H$-Free Colourings of Random Graphs

Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $\lambda(H,\ell)$ for the rainbow entropy--Tur\'an density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $\lambda(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy--Tur\'an parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.

math.CO↗

arXiv2026

Convex Transference for Degree Powers in Extremal Set Systems

For a family $\mathcal{F}\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal{F}}(R)=|\{F\in\mathcal{F}:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal{F})=\sum_{R\in\binom{[n]}r}d_{\mathcal{F}}(R)^p$; at the codegree level, write $co_p(\mathcal{F})=\ell_{k-1,p}(\mathcal{F})$. We introduce a new convex-transference method for degree-power extremal problems and develop it into a reusable input--transfer--rigidity framework independent of any particular set-system problem. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the Wu--Zhang quadratic theorem to real exponents and answers a problem of Zhou--Yuan throughout the sharp Erd\H{o}s--Ko--Rado range. Second, if $n\geq2k$, a full point-star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with complete equality classification; thus the framework is not confined to codegrees. Third, if $\nu(\mathcal{F})\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal{F})$ is uniquely maximized, up to isomorphism, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or nonexplicit sufficiently-large assumptions with an explicit linear range valid for arbitrary uniformity.

math.CO↗

arXiv2026

Projective Ore-Degree Conditions for Intersection Theorems in Vector Spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The \emph{projective Ore-degree} of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. The Ore--Erd\H{o}s--Ko--Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore--Hilton--Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.

math.CO↗

arXiv2026

Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs

Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge of $G$ independently with probability $p$. Put $q:=1-p$. Motivated by a conjecture of Bedert, Dragani\'c, M\"uyesser, and Pavez-Sign\'e on Hamilton cycles in percolated Cayley graphs, we establish the corresponding matching and $2$-factor statements uniformly over the larger class of all connected vertex-transitive host graphs. For every $A>0$, if $q^d\le n^{-(5A+250)},$ then, with probability at least $1-n^{-A}$, the graph $G_p$ has a perfect matching when $n$ is even and is factor-critical when $n$ is odd. Separately, if $0<\epsilon<1$ and $ \epsilon^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-\epsilon)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ \nu_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $\nu_2(G)$ is the maximum number of pairwise edge-disjoint spanning 2-factors in $G$. Thus logarithmic-order percolation already forces these two factor-theoretic consequences of Hamiltonicity beyond the Cayley setting.

math.CO↗

arXiv2026

Degree-restricted semi-saturation numbers of cliques and its applications

A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. The semi-saturation number $ssat(n,F)$ is the minimum number of edges in an $F$-semi-saturated graph of order $n$. In this paper we investigate the semi-saturation number of $K_r$ on $n$ vertices with maximal degree at most $\Delta$, denoted by $ssat^{\Delta}(n,K_r)$. This investigation was suggested by Erd\H os, R\'enyi and S\'os, who in 1966 considered the graph of diameter 2 with degree restrictions, equivalently $ssat^{\Delta}(n,K_3)$. The following are some of our results. For arbitrary $r \geq 4$, we show that the limit $ \lim_{n \rightarrow \infty} ssat^{cn}(n,K_r)/n$ exists for all $0 < c \leq 1$, except for some sparse values of $c$ contained in a countable and rational sequence $c_i \rightarrow 0$. Moreover, we establish the asymptotic behaviour of this limit for $\frac{r}{r+2} < c <1$ and determine the exact value of $ssat^{\Delta}(n,K_r)$ for some specific $\Delta$. As an application, we determine the relation between the saturation number of the join graph $K_r \vee F$ and that of $F$ for a large class of pairs $(r,F)$.

math.CO↗

arXiv2026

All minimum $C_4$-saturated multipartite graphs

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part. In this paper, we determine $sat(K_4^n,C_4)$ for all $ n \geq 2$. Moreover, we determine all extremal configurations of $sat(K_k^n,C_4)$ for all $n\ge 2$ and $k\ge 4 $.

math.CO↗

arXiv2026

On degree bounds of $k$-uniform hypergraphs with bounded matching number

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1(\mathcal{F})\ge d_2(\mathcal{F})\ge\cdots\ge d_n(\mathcal{F})$ be the vertex degrees arranged in non-increasing order. For integers $k\ge 2$, $s\ge 2$ and $n > 2sk$, we prove that if the $2sk$-th largest degree satisfies $d_{2sk} > \binom{n-1}{k-1} - \binom{n-s}{k-1},$ then $\mathcal{F}$ contains a matching of size at least $s$. Moreover, by relaxing the range of $n$, we obtain the same bound for the $(k+2s-2)$-th largest degree vertex. Note that the number $k+2s-2$ is optimal. For a $k$-set of vertices $S \subseteq [n]$, the degree of $S$ is defined as $\deg(S) = \sum_{v \in S} \deg(v)$, and the minimum of $\deg(S)$ over all non-edge $k$-subsets $S \notin E(\mathcal{F})$ of $V(\mathcal{F})$ is the \textit{Ore-degree} of $\mathcal{F}$, denoted by $\sigma_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $\sigma_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ then $\mathcal{F}$ contains a matching of size $s$. They also conjectured that the result holds when $n > sk$. As a corollary, we prove that the bound on $n$ can be taken to be linear in $sk$ ($ n \geq 3sk $).

math.CO↗

arXiv2026

Algorithm for finding vertex-edge domination number on graphs with bounded treewidth and related problems on planar graphs

Given a graph $G=(V,E)$, a vertex $u \in V$ {\em ve-dominates} all edges incident to any vertex of $N_G[u]$. A set $S \subseteq V$ is a {\em ve-dominating set} if for all edges $e\in E$, there exists a vertex $u\in S$ such that $u$ ve-dominates $e$. The minimum cardinality among all ve-dominating sets is known as the \textit{vertex-edge domination number} (or simply ve-domination number) and denoted by $\gamma_{ve}(G)$. Finding a minimum ve-dominating set was proved to be NP-complete. Restricted to trees, the problem admits a linear-time algorithm. Treewidth is a commonly used parameter for solving NP-hard problems. In this paper, we present a polynomial-time algorithm for finding a minimum ve-dominating set on graphs with bounded treewidth. Moreover, we show that the treewidth of a planar graph $G$ with ve-domination number $\gamma_{ve}(G)$ is $O(\sqrt{\gamma_{ve}(G)})$ and present an $O(c^{\sqrt{k}}|V(G)|)$-time algorithm for the $k$-ve-domination problem on planar graphs.

math.CO↗

arXiv2026

A generalization of Erd\H{o}s-Hajnal problem on paths with equal-degree endpoints

Erd\H{o}s and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture.

math.CO↗

arXiv2026

A Quadratic Vertex Threshold for Isolated Cliques in the Minimum Degree Kruskal-Katona Problem for 3-Uniform Hypergraphs

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal-Katona theorem states that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_{k-1}\mathcal{F}|\geq\binom{t}{k-1}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_{k-1}\mathcal{F}|$ be? In this article, for the case $k=3$, we prove that, for every sufficiently large integer $t$, every extremal hypergraph for this problem contains an isolated copy of $K_{t+1}^3$ whenever $|X| \geq ct^2 + o(t^2)$, with the constant $c = 1 + \sqrt{928/33}$. Our proof uses a graph transformation that regularizes the neighborhood structure of extremal graphs, reducing the problem to a counting argument on the neighbors of a disjoint clique family. This gives a quadratic-order threshold for the every-extremal version of the problem, compared with the cubic-order threshold of F\"{u}redi and Zhao [SIAM J.\ Discrete Math.\ 36(4), 2022].

math.CO↗

arXiv2026

Shadows of Uniform Hypergraphs under a Minimum Degree Condition

Given a set $X$ and an integer $t$, let $\mathcal{F}$ be a family of $k$-subsets of $X$. The Kruskal--Katona theorem implies that if $|\mathcal{F}|\geq \binom{t}{k}$, then $|\partial_\ell\mathcal{F}|\geq\binom{t}{\ell}$. The minimum degree version of this problem asks: if $\delta(\mathcal{F})\geq \binom{t}{k-1}$, how small can $|\partial_\ell\mathcal{F}|$ be? We call a hypergraph \textit{extremal} if it achieves the minimum value of $|\partial_\ell \mathcal{F}|$ subject to the degree condition $\delta(\mathcal{F}) \geq \binom{t}{k-1}$. F\"uredi and Zhao [SIAM J. Discrete Math. 36(4), 2022] proved that for $k=3$, $\ell=2$ and $t\ge 2$, every extremal hypergraph contains an isolated copy of $K_{t+1}^3$ when $|X| > \frac{1}{4}(t+1)^2(t+2)$. In this article, we study the general case $k > \ell \geq 2$. By developing a hypergraph transformation that combines shifting operations with antilexicographic compression, we prove that, for every integer $t\ge k-1$, there exists an extremal hypergraph containing an isolated copy of $K^{k}_{t+1}$ whenever $|X| > \frac{1}{4}(t+1)^2\binom{t-1}{\ell-2} + 3t+1$. In the case when $k=3$ and $\ell=2$, this gives the threshold $\frac14(t+1)^2+3t+1$, which is smaller than $\frac14(t+1)^2(t+2)$ for every $t\ge3$; for $t=2$, the two thresholds give the same integer condition on $|X|$.

math.CO↗

arXiv2026

Counting sunflowers with restricted matching number

For $\mathcal H\subseteq\binom{[n]}k$, let $\nu(\mathcal H)$ denote its matching number and let $d_{\mathcal H}(E)$ denote the codegree of a $(k-1)$-set $E$. We study the codegree moments $ co_p(\mathcal H)=\sum_{E\in\binom{[n]}{k-1}}d_{\mathcal H}(E)^p $ and the number of copies of the $k$-uniform $l$-petal sunflower $S_{k,l}^{k-1}$ whose core has size $k-1$. For fixed $k,s,l$ and $p$, and all sufficiently large $n$, we prove that among all $k$-uniform families with matching number at most $s$, both quantities are uniquely maximized by \[ \mathcal H_{n,k,s}=\left\{F\in\binom{[n]}k:F\cap[s]\neq\emptyset\right\}. \] Thus the sunflower result is the generalized Tur\'an problem $\operatorname{ex}_k(n,S_{k,l}^{k-1},M_{s+1})$. When $k=3$, we make the threshold effective: for every integer $p\geq1$, the codegree-moment conclusion holds for $n\geq6s+3$. The proof combines a Hilton--Milner-type stability theorem of Frankl and Kupavskii with a high-codegree-shadow reduction and a convex moment estimate.

math.CO↗

arXiv2025

Extremal results on Berge disjoint paths

The well-known Erd\H{o}s-Gallai Theorem gave the Tur\'an number of paths. Bushaw and Kettle generalized this result to consider the Tur\'an number of disjoint paths. Since then, many studies are focused on the Tur\'an number of linear forest. For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-} F$ if there is a bijection $\phi: E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-} F$ a Berge path. The Tur\'an number of Berge paths was initially studied by Gy\H{o}ri, Katona and Lemons. They gave the value of $\text{ex}_r(n,\text{Berge-}P_\ell)$ for $\ell>r+1$. This result is a generalization of Erd\H{o}s-Galli Theorem. Since then, the Tur\'an number of Berge paths has received widespread attention. Recently, Zhou, Gerbner and Yuan initially studied the Tur\'an number of Berge disjoint paths and for the cases when all the paths have odd length. In this paper, we give a more general result, which gives the exact value of $\mathrm{ex}_r(n,\text{Berge-} kP_{\ell})$ for all $k\geq 2$, $r\ge 3$, and $\ell\geq r+7$.

math.CO↗

arXiv2025

The maximum sum of sizes of non-empty cross $L$-intersecting families

Let $n$, $r$, and $k$ be positive integers such that $k, r \geq 2$, $L$ a non-empty subset of $[k]$, and $\mathcal{F}_i \subseteq \binom{[n]}{k}$ for $1 \leq i \leq r$. We say that non-empty families $\mathcal{F}_1, \mathcal{F}_2, \ldots, \mathcal{F}_r$ are $r$-cross $L$-intersecting if $\left| \bigcap_{i=1}^r F_i \right| \in L$ for every choice of $F_i \in \mathcal{F}_i$ with $1 \leq i \leq r$. They are called pairwise cross $L$-intersecting if $|A \cap B| \in L$ for all $A \in \mathcal{F}_i$, $B \in \mathcal{F}_j$ with $i \neq j$. If $r=2$, we simply say cross $L$-intersecting instead of $2$-cross $L$-intersecting or pairwise cross $L$-intersecting. In this paper, we determine the maximum possible sum of sizes of non-empty cross $L$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ for all admissible $n$, $k$, and $L$, and we characterize all the extremal structures. We also establish the maximum value of the sum of sizes of families $\mathcal{F}_1, \dots, \mathcal{F}_r$ that are both pairwise cross $L$-intersecting and $r$-cross $L$-intersecting, provided $n$ is sufficiently large and $L$ satisfies certain conditions. Furthermore, we characterize all such families attaining the maximum total size.

math.CO↗

arXiv2025

Edge version of the inducibility via the entropy method

The inducibility of a graph $H$ is about the maximum number of induced copies of $H$ in a graph on $n$ vertices. We consider its edge version, that is, the maximum number of induced copies of $H$ in a graph with $m$ edges. Let $c(G,H)$ be the number of induced copies of $H$ in $G$ and $\rho(H,m) = \max \{c(G,H) \mid |E(G)| = m\}$. For any graph $H$, we prove that $\rho(H,m) = \Theta(m^{\alpha_f(H)})$ where $\alpha_f(H)$ is the fractional independence number of $H$. Therefore, we now focus on the constant factor in front of $m^{\alpha_f(H)}$. In this paper, we give some results of $\rho(H,m)$ when $H$ is a cycle or path. We conjecture that for any cycle $C_k$ with $k \ge 5$, $\rho(C_k,m)= (1+o(1))\left( m/k\right)^{k/2}$ and the bound achieves by the blow up of $C_k$. For even cycles, we establish an upper bound with an extra constant factor. For odd cycles, we can only establish an upper bound with an extra factor depending on $k$. We prove that $\rho(P_{2l},m) \le \frac{m^l}{2(l-1)^{l-1}}$ and $\rho(P_{2l+1},m) \le \frac{m^{l+1}}{4l^l}$, where $l \ge 2$. We also conjecture the asymptotic value of $\rho(P_k, m)$. The entropy method is mainly used to prove our results.

math.CO↗

arXiv2025

Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs

Let $G$ be a graph on $n$ vertices and $t$ an integer. The reconfiguration graph of $G$, denoted by $R_t(G)$, consists of all $t$-colorings of $G$ and two $t$-colorings are adjacent if they differ on exactly one vertex. The $t$-recoloring diameter of $G$ is the diameter of $R_t(G)$. For a $d$-degenerate graph $G$, $R_t(G)$ is connected when $t \ge d+2$~(Dyer et al., 2006). Furthermore, the $t$-recoloring diameter is $O(n^2)$ when $t \ge 3(d+1)/2$~(Bousquet et al., 2022), and it is $O(n)$ when $t \ge 2d+2$~(Bousquet and Perarnau, 2016). For a $d$-degenerate and chordal graph $G$, the $t$-recoloring diameter of $G$ is $O(n^2)$ when $t \ge d+2$~(Bonamy et al. 2014). If $G$ is a graph of treewidth at most $k$, then $G$ is also $k$-degenerate, and the previous results hold. Moreover, when $t \ge k+2$, the $t$-recoloring diameter is $O(n^2)$~(Bonamy and Bousquet, 2013). When $k=2$, the $t$-recoloring diameter of $G$ is linear when $t \ge 5$~(Bartier, Bousquet and Heinrich, 2021) and the result is tight. In this paper, we prove that if $G$ is $d$-degenerate and chordal, then the $t$-recoloring diameter of $G$ is $O(n)$ when $t \ge 2d+1$. Moreover, if the treewidth of $G$ is at most $k$, then the $t$-recoloring diameter is $O(n)$ when $t \ge 2k+1$. This result is a generalization of the previous results on graphs of treewidth at most two.

math.CO↗