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Zhen He

Publications and source records attributed to Zhen He.

At least 19 recordsLinked to original sources
arXiv2026

A note on the saturation number for unions of three cliques

A graph $G$ is $F$-saturated if $G$ contains no copy of $F$ but $G+e$ contains a copy of $F$ for every missing edge $e$ of $G$. The saturation number $\sat(n,F)$ is the minimum number of edges in an $n$-vertex $F$-saturated graph. Motivated by a problem posed by Faudree, Ferrara, Gould, and Jacobson concerning $K_p\cup K_q\cup K_{q+1}$, we determine the saturation number and the unique extremal graph for $K_p\cup K_q\cup K_r$ whenever $2\le p\le q<r<p+q$ and $n$ is sufficiently large. Together with the previously known results for $r\ge p+q$ and for $r=q$, this completes the determination of the saturation number and the extremal graphs for unions of three cliques, for all sufficiently large $n$.

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arXiv2026

The maximum number of odd cycles in planar graphs forbidding shorter odd cycles

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n, H, \mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex planar graph that contains no graph $F \in \mathcal{F}$ as a subgraph. When only induced copies of $H$ are counted, we denote the corresponding generalized planar Tur\'an number by $\mathrm{ex}_\mathcal{P}(n, H^{\mathrm{ind}}, \mathcal{F})$. Gy\H{o}ri and Karim determined $\mathrm{ex}_\mathcal{P}(n, C_{5}, \{C_3\})$. In this paper, we determine the exact value of $\mathrm{ex}_\mathcal{P}(n, C_{2k+1}, \{C_3,C_5,\ldots,C_{2k-1}\})$ for every $k \ge 3$. Since all shorter odd cycles are forbidden, every $C_{2k+1}$ is induced. This problem is closely related to the inducibility of odd cycles in planar graphs. Ghosh, Gy\H{o}ri, Janzer, Paulos, Salia and Zamora~(and independently Savery) determined the exact value of $\mathrm{ex}_\mathcal{P}(n, C_5^{\mathrm{ind}}, \emptyset)$. Moreover, they established a conjecture for all odd cycles $C_{2k+1}$ with $k \ge 3$. Our result confirms their conjecture under the additional assumption that all shorter odd cycles are forbidden.

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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 $(r,F)$.

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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 $.

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arXiv2026

Tur\'an-Type Extremal Results for Distance-$k$ Graphs

We study Tur\'an-type extremal problems for distance graphs, motivated by work of Csikv\'ari, Bollob\'as, Tyomkyn, and Uzzell. We determine the maximum number of vertex pairs at distance three in an $n$-vertex graph with no triangle formed by these pairs, resolving the first case of a conjecture of Tyomkyn and Uzzell. We also determine the maximum number of vertex pairs at distance two in an $n$-vertex graph with no triangle formed by these pairs and give a complete characterization of the extremal graphs, settling another problem of Tyomkyn and Uzzell.

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arXiv2025

The Connected Bipartite Tur\'an Problem for Long Cycles and Paths

Caro, Patk\'os, and Tuza initiated a systematic study of the bipartite Tur\'an number for trees, and in particular asked for the extremal number of edges in connected bipartite graphs with prescribed color-class sizes that contain no paths of given lengths. In this paper, we determine these numbers exactly and describe all corresponding extremal configurations. Our approach first establishes a more general result for long cycles: we determine the exact structure of all 2-connected bipartite graphs with no cycle of length at least a given constant. The proof combines Kopylov's method for long cycles with a strengthened version of Jackson's classical lemma, in which every extremal configuration is characterized. To highlight the applicability of our results, we conclude with applications yielding concise proofs of classical theorems on bipartite Tur\'an numbers, notably rederiving the results of Gy\'arf\'as, Rousseau, and Schelp for paths and Jackson for long cycles.

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arXiv2025

OCELOT 2023: Cell Detection from Cell-Tissue Interaction Challenge

Pathologists routinely alternate between different magnifications when examining Whole-Slide Images, allowing them to evaluate both broad tissue morphology and intricate cellular details to form comprehensive diagnoses. However, existing deep learning-based cell detection models struggle to replicate these behaviors and learn the interdependent semantics between structures at different magnifications. A key barrier in the field is the lack of datasets with multi-scale overlapping cell and tissue annotations. The OCELOT 2023 challenge was initiated to gather insights from the community to validate the hypothesis that understanding cell and tissue (cell-tissue) interactions is crucial for achieving human-level performance, and to accelerate the research in this field. The challenge dataset includes overlapping cell detection and tissue segmentation annotations from six organs, comprising 673 pairs sourced from 306 The Cancer Genome Atlas (TCGA) Whole-Slide Images with hematoxylin and eosin staining, divided into training, validation, and test subsets. Participants presented models that significantly enhanced the understanding of cell-tissue relationships. Top entries achieved up to a 7.99 increase in F1-score on the test set compared to the baseline cell-only model that did not incorporate cell-tissue relationships. This is a substantial improvement in performance over traditional cell-only detection methods, demonstrating the need for incorporating multi-scale semantics into the models. This paper provides a comparative analysis of the methods used by participants, highlighting innovative strategies implemented in the OCELOT 2023 challenge.

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arXiv2025

Sets avoiding a rainbow solution to the generalized Schur equation

A classical result in combinatorial number theory states that the largest subset of $[n]$ avoiding a solution to the equation $x+y=z$ is of size $\lceil n/2 \rceil$. For all integers $k>m$, we prove multicolored extensions of this result where we maximize the sum and product of the sizes of sets $A_1,A_2,\dots,A_k \subseteq [n]$ avoiding a rainbow solution to the Schur equation $x_1+x_2+\dots+x_m=x_{m+1}$. Moreover, we determine all the extremal families.

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arXiv2025

The saturation number of wheels

A graph $G$ is said to be $F$-free, if $G$ does not contain any copy of $F$. $G$ 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 $G+e$. $G$ is said to be $F$-saturated, if $G$ is $F$-free and $F$-semi-saturated. The saturation number $sat(n,F)$ (resp. semi-saturation number $ssat(n,F)$) is the minimum number of edges in an $F$-saturated (resp. $F$-semi-saturated) graph of order $n$. In this paper we proved several results on the (semi)-saturation number of the wheel graph $W_k=K_1 \vee C_k$. Let $k,n$ be positive integers with $k \geq 8$ and $n \geq 56k^3$, we showed that $(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k)$. We also establish the lower bound of semi-saturation number of $W_k$ with restriction on maximum degree.

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arXiv2025

Bridging the Vision-Brain Gap with an Uncertainty-Aware Blur Prior

Can our brain signals faithfully reflect the original visual stimuli, even including high-frequency details? Although human perceptual and cognitive capacities enable us to process and remember visual information, these abilities are constrained by several factors, such as limited attentional resources and the finite capacity of visual memory. When visual stimuli are processed by human visual system into brain signals, some information is inevitably lost, leading to a discrepancy known as the \textbf{System GAP}. Additionally, perceptual and cognitive dynamics, along with technical noise in signal acquisition, degrade the fidelity of brain signals relative to the visual stimuli, known as the \textbf{Random GAP}. When encoded brain representations are directly aligned with the corresponding pretrained image features, the System GAP and Random GAP between paired data challenge the model, requiring it to bridge these gaps. However, in the context of limited paired data, these gaps are difficult for the model to learn, leading to overfitting and poor generalization to new data. To address these GAPs, we propose a simple yet effective approach called the \textbf{Uncertainty-aware Blur Prior (UBP)}. It estimates the uncertainty within the paired data, reflecting the mismatch between brain signals and visual stimuli. Based on this uncertainty, UBP dynamically blurs the high-frequency details of the original images, reducing the impact of the mismatch and improving alignment. Our method achieves a top-1 accuracy of \textbf{50.9\%} and a top-5 accuracy of \textbf{79.7\%} on the zero-shot brain-to-image retrieval task, surpassing previous state-of-the-art methods by margins of \textbf{13.7\%} and \textbf{9.8\%}, respectively. Code is available at \href{https://github.com/HaitaoWuTJU/Uncertainty-aware-Blur-Prior}{GitHub}.

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arXiv2025

The Rainbow Saturation Number of Cycles

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$ does not contain a rainbow copy of $F$, but the addition of any nonedge with any color on it would create a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in an $F$-rainbow saturated graph with order $n$. In this paper we proved several results on cycle rainbow saturation. For $n \geq 5$, we determined the exact value of $rsat(n,C_4)$. For $ n \geq 15$, we proved that $\frac{3}{2}n-\frac{5}{2} \leq rsat(n,C_{5}) \leq 2n-6$. For $r \geq 6$ and $n \geq r+3$, we showed that $ \frac{6}{5}n \leq rsat(n,C_r) \leq 2n+O(r^2)$. Moreover, we establish better lower bound on $C_r$-rainbow saturated graph $G$ while $G$ is rainbow.

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arXiv2024

Partite saturation number of cycles

A graph $H$ 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 containing $n$ vertices in each part and $C_\ell$ be the cycle of length $\ell$. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_\ell)$ for all $ \ell \geq 4, k \geq 2$ except $(\ell,k)=(4,4)$. Moreover, we determined the exact value of $sat(K_k^n,C_\ell)$ for $ k>\ell=4 $ and $5 \geq \ell>k \geq 3$ and $(\ell,k)=(6,2)$.

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arXiv2024

Off to new Shores: A Dataset & Benchmark for (near-)coastal Flood Inundation Forecasting

Floods are among the most common and devastating natural hazards, imposing immense costs on our society and economy due to their disastrous consequences. Recent progress in weather prediction and spaceborne flood mapping demonstrated the feasibility of anticipating extreme events and reliably detecting their catastrophic effects afterwards. However, these efforts are rarely linked to one another and there is a critical lack of datasets and benchmarks to enable the direct forecasting of flood extent. To resolve this issue, we curate a novel dataset enabling a timely prediction of flood extent. Furthermore, we provide a representative evaluation of state-of-the-art methods, structured into two benchmark tracks for forecasting flood inundation maps i) in general and ii) focused on coastal regions. Altogether, our dataset and benchmark provide a comprehensive platform for evaluating flood forecasts, enabling future solutions for this critical challenge. Data, code & models are shared at https://github.com/Multihuntr/GFF under a CC0 license.

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arXiv2024

The saturation number for unions of four cliques

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \cup K_{p_2} \cup K_{p_3})$ with $p_3 \ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \cup K_{p_2} \cup K_{p_3} \cup K_{p_4})$ with $p_{i+1} - p_i \ge p_1$ for $2 \le i\le 3$ and $4\le p_1\le p_2$ is determined.

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arXiv2024

Saturation Numbers for Linear Forests $P_7+tP_2$

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. In this paper, we determine the saturation number sat$(n,P_7+tP_2)$ for $n\geq \frac {14}{5}t+27$ and characterize the extremal graphs for $n\geq \frac{14}{13}(3t+25)$.

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arXiv2024

Minimum saturated graphs for unions of cliques

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t-1)K_q)$ for every $2\le p\le q$ and $t\ge 2$.

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arXiv2024

Group Activity Recognition using Unreliable Tracked Pose

Group activity recognition in video is a complex task due to the need for a model to recognise the actions of all individuals in the video and their complex interactions. Recent studies propose that optimal performance is achieved by individually tracking each person and subsequently inputting the sequence of poses or cropped images/optical flow into a model. This helps the model to recognise what actions each person is performing before they are merged to arrive at the group action class. However, all previous models are highly reliant on high quality tracking and have only been evaluated using ground truth tracking information. In practice it is almost impossible to achieve highly reliable tracking information for all individuals in a group activity video. We introduce an innovative deep learning-based group activity recognition approach called Rendered Pose based Group Activity Recognition System (RePGARS) which is designed to be tolerant of unreliable tracking and pose information. Experimental results confirm that RePGARS outperforms all existing group activity recognition algorithms tested which do not use ground truth detection and tracking information.

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