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In this paper, extending a result of Brooks et.al. [arXiv:2403.04110], we show that if an outerplanar graph $G$ with minimum degree at least $2$ has positive Lin-Lu-Yau curvature on every vertex pair, then $G$ has at most $10$ vertices, and this upper bound is sharp., Comment: 17 pages

Erd\H{o}s, Pach, Pollack, and Tuza [J. Combin. Theory Ser. B, 47(1) (1989), 73--79] proved that the diameter of a connected $n$-vertex graph with minimum degree $\delta$ is at most $\frac{3n}{\delta+1}+O(1)$. The oriented diameter of an undirected graph $G$, denoted by $\overrightarrow{diam}(G)$, is the minimum diameter of a strongly connected orientation of $G$. Bau and Dankelmann [European J. Combin., 49 (2015), 126--133] showed that for every bridgeless $n$-vertex graph $G$ with minimum degree $\delta$, $\overrightarrow{diam}(G) \leq \frac{11n}{\delta+1}+9$. They also showed an infinite family of graphs with oriented diameter at least $\frac{3n}{\delta+1} + O(1)$ and posed the problem of determining the smallest possible value $c$ for which $\overrightarrow{diam}(G) \leq c \cdot\frac{3n}{\delta+1}+O(1)$ holds. In this paper, we show that the smallest value $c$ such that the upper bound above holds for all $\delta\geq 2$ is $1$, which is best possible., Comment: 12 pages

Existing camouflaged object detection~(COD) methods depend heavily on large-scale pixel-level annotations.However, acquiring such annotations is laborious due to the inherent camouflage characteristics of the objects.Semi-supervised learning offers a promising solution to this challenge.Yet, its application in COD is hindered by significant pseudo-label noise, both pixel-level and instance-level.We introduce CamoTeacher, a novel semi-supervised COD framework, utilizing Dual-Rotation Consistency Learning~(DRCL) to effectively address these noise issues.Specifically, DRCL minimizes pseudo-label noise by leveraging rotation views' consistency in pixel-level and instance-level.First, it employs Pixel-wise Consistency Learning~(PCL) to deal with pixel-level noise by reweighting the different parts within the pseudo-label.Second, Instance-wise Consistency Learning~(ICL) is used to adjust weights for pseudo-labels, which handles instance-level noise.Extensive experiments on four COD benchmark datasets demonstrate that the proposed CamoTeacher not only achieves state-of-the-art compared with semi-supervised learning methods, but also rivals established fully-supervised learning methods.Our code will be available soon., Comment: Accepted to ECCV 2024

The Agriculture-Vision Challenge at CVPR 2024 aims at leveraging semantic segmentation models to produce pixel level semantic segmentation labels within regions of interest for multi-modality satellite images. It is one of the most famous and competitive challenges for global researchers to break the boundary between computer vision and agriculture sectors. However, there is a serious class imbalance problem in the agriculture-vision dataset, which hinders the semantic segmentation performance. To solve this problem, firstly, we propose a mosaic data augmentation with a rare class sampling strategy to enrich long-tail class samples. Secondly, we employ an adaptive class weight scheme to suppress the contribution of the common classes while increasing the ones of rare classes. Thirdly, we propose a probability post-process to increase the predicted value of the rare classes. Our methodology achieved a mean Intersection over Union (mIoU) score of 0.547 on the test set, securing second place in this challenge.

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\geq s \geq 2$, there is an integer $\xi_{t}$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+\xi_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $\xi_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2212.05540

For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if $\alpha(G-S)\geq \alpha(G)-\ell$ for every $S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J. Discrete Math., 36 (2022) 229--240] shows that every $(k,\ell)$-stable graph $G$ satisfies $\alpha(G) \le \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$. A $(k,\ell)$-stable graph $G$ is tight if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$; and $q$-tight for some integer $q\ge0$ if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell-q$. In this paper, we first prove that for all $k\geq 24$, the only tight $(k, 0)$-stable graphs are $K_{k+1}$ and $K_{k+2}$, answering a question of Dong and Luo [arXiv: 2401.16639]. We then prove that for all nonnegative integers $k, \ell, q$ with $k\geq 3\ell+3$, every $q$-tight $(k,\ell)$-stable graph has at most $k-3\ell-3+2^{3(\ell+2q+4)^2}$ vertices, answering a question of Dong and Luo in the negative., Comment: 11 pages

Interfacial ferroelectricity emerges in heterostructures consisting of nonpolar van der Waals (vdW) layers, greatly expanding the scope of two dimensional ferroelectrics. In particular, the unconventional moire ferroelectricity observed in bilayer graphene/boron nitride (BN) heterostructures, exhibits promising functionalities with topological current, superconductivity and synaptic responses. However, the debate about its mechanism - correlation driven charge transfer between two graphene layers - limits device reproducibility and hence large-scale production. Here by designing a single-layer graphene encapsulated by lattice-mismatched WSe2, we identify the ferroelectricity as stemming from - instead of graphene moire bands - the particular BN, where interfacial sliding ferroelectricity must play a role. With similar structures, multilayer twisted MoS2 is found to reproduce the ferroelectricity. The key is a conductive moire ferroelectric, where the screened gate and the pinned domain wall together result in unchanged electronic states, i.e. anomalous screening. The intimate connection to interfacial sliding ferroelectricity thus provides advantages of diverse choices of constituent materials and robust polarization switching while preserving the unique anomalous screening, paving the way to reproducible and reliable memory-based devices in artificial intelligence.

In this paper, we show that all simple outerplanar graphs $G$ with minimum degree at least $2$ and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most $9$. Furthermore, if $G$ is maximally outerplanar, then $G$ has at most $10$ vertices. Both upper bounds are sharp., Comment: 10 pages

In this paper, we show that the oriented diameter of any $n$-vertex $2$-connected near triangulation is at most $\lceil{\frac{n}{2}}\rceil$ (except for seven small exceptions), and the upper bound is tight. This extends a result of Wang et.al. on the oriented diameter of maximal outerplanar graphs, and improves an upper bound of $n/2+O(\sqrt{n})$ on the oriented diameter of planar triangulations by Mondal, Parthiban and Rajasingh.

Edge/fog computing, as a distributed computing paradigm, satisfies the low-latency requirements of ever-increasing number of IoT applications and has become the mainstream computing paradigm behind IoT applications. However, because large number of IoT applications require execution on the edge/fog resources, the servers may be overloaded. Hence, it may disrupt the edge/fog servers and also negatively affect IoT applications' response time. Moreover, many IoT applications are composed of dependent components incurring extra constraints for their execution. Besides, edge/fog computing environments and IoT applications are inherently dynamic and stochastic. Thus, efficient and adaptive scheduling of IoT applications in heterogeneous edge/fog computing environments is of paramount importance. However, limited computational resources on edge/fog servers imposes an extra burden for applying optimal but computationally demanding techniques. To overcome these challenges, we propose a Deep Reinforcement Learning-based IoT application Scheduling algorithm, called DRLIS to adaptively and efficiently optimize the response time of heterogeneous IoT applications and balance the load of the edge/fog servers. We implemented DRLIS as a practical scheduler in the FogBus2 function-as-a-service framework for creating an edge-fog-cloud integrated serverless computing environment. Results obtained from extensive experiments show that DRLIS significantly reduces the execution cost of IoT applications by up to 55%, 37%, and 50% in terms of load balancing, response time, and weighted cost, respectively, compared with metaheuristic algorithms and other reinforcement learning techniques.

Fine-tuning visual models has been widely shown promising performance on many downstream visual tasks. With the surprising development of pre-trained visual foundation models, visual tuning jumped out of the standard modus operandi that fine-tunes the whole pre-trained model or just the fully connected layer. Instead, recent advances can achieve superior performance than full-tuning the whole pre-trained parameters by updating far fewer parameters, enabling edge devices and downstream applications to reuse the increasingly large foundation models deployed on the cloud. With the aim of helping researchers get the full picture and future directions of visual tuning, this survey characterizes a large and thoughtful selection of recent works, providing a systematic and comprehensive overview of existing work and models. Specifically, it provides a detailed background of visual tuning and categorizes recent visual tuning techniques into five groups: prompt tuning, adapter tuning, parameter tuning, and remapping tuning. Meanwhile, it offers some exciting research directions for prospective pre-training and various interactions in visual tuning., Comment: 37 pages. Accepted to ACM CSUR

An accurate and explainable automatic monitoring system is critical for the safety of high efficiency energy conversion plants that operate under extreme working condition. Nonetheless, currently available data-driven monitoring systems often fall short in meeting the requirements for either high-accuracy or interpretability, which hinders their application in practice. To overcome this limitation, a data-driven approach, AttentionMixer, is proposed under a generalized message passing framework, with the goal of establishing an accurate and interpretable radiation monitoring framework for energy conversion plants. To improve the model accuracy, the first technical contribution involves the development of spatial and temporal adaptive message passing blocks, which enable the capture of spatial and temporal correlations, respectively; the two blocks are cascaded through a mixing operator. To enhance the model interpretability, the second technical contribution involves the implementation of a sparse message passing regularizer, which eliminates spurious and noisy message passing routes. The effectiveness of the AttentionMixer approach is validated through extensive evaluations on a monitoring benchmark collected from the national radiation monitoring network for nuclear power plants, resulting in enhanced monitoring accuracy and interpretability in practice.

Parameter-efficient transfer learning (PETL) is an emerging research spot aimed at inexpensively adapting large-scale pre-trained models to downstream tasks. Recent advances have achieved great success in saving storage costs for various pre-trained models by updating a small number of parameters instead of full tuning. However, we notice that most existing PETL methods still incur non-negligible latency during inference. In this paper, we propose a parameter-efficient and computational friendly adapter for giant vision models, called RepAdapter. Specifically, we first prove that common adaptation modules can also be seamlessly integrated into most giant vision models via our structural re-parameterization, thereby achieving zero-cost during inference. We then investigate the sparse design and effective placement of adapter structure, helping our RepAdaper obtain other advantages in terms of parameter efficiency and performance. To validate RepAdapter, we conduct extensive experiments on 27 benchmark datasets of three vision tasks, i.e., image and video classifications and semantic segmentation. Experimental results show the superior performance and efficiency of RepAdapter than the state-of-the-art PETL methods. For instance, RepAdapter outperforms full tuning by +7.2% on average and saves up to 25% training time, 20% GPU memory, and 94.6% storage cost of ViT-B/16 on VTAB-1k. The generalization ability of RepAdapter is also well validated by a bunch of vision models. Our source code is released at https://github.com/luogen1996/RepAdapter.

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there is an integer $\xi_t$ such that the maximum spread of an $n$-vertex $K_{2,t}$-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor$ copies of $K_t$ and $n-1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. The extremal graph is unique, except when $t\equiv 4 \mod 12$ and $\frac{2n+ \xi_t} {3t}$ is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+\xi_t}{3t}\rfloor-1$ copies of $K_t$ and $n-1-t(\lfloor \frac{2n+\xi_t}{3t}\rfloor-1)$ isolated vertices. Furthermore, we give an explicit formula for $\xi_t$., Comment: Minor revisions. arXiv admin note: text overlap with arXiv:2209.13776

Enterprise relation extraction aims to detect pairs of enterprise entities and identify the business relations between them from unstructured or semi-structured text data, and it is crucial for several real-world applications such as risk analysis, rating research and supply chain security. However, previous work mainly focuses on getting attribute information about enterprises like personnel and corporate business, and pays little attention to enterprise relation extraction. To encourage further progress in the research, we introduce the CEntRE, a new dataset constructed from publicly available business news data with careful human annotation and intelligent data processing. Extensive experiments on CEntRE with six excellent models demonstrate the challenges of our proposed dataset.

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $\lceil(2n-2)/3\rceil$ vertices and $\lfloor(n-4)/3\rfloor$ isolated vertices.

Resource management is the principal factor to fully utilize the potential of Edge/Fog computing to execute real-time and critical IoT applications. Although some resource management frameworks exist, the majority are not designed based on distributed containerized components. Hence, they are not suitable for highly distributed and heterogeneous computing environments. Containerized resource management frameworks such as FogBus2 enable efficient distribution of framework's components alongside IoT applications' components. However, the management, deployment, health-check, and scalability of a large number of containers are challenging issues. To orchestrate a multitude of containers, several orchestration tools are developed. But, many of these orchestration tools are heavy-weight and have a high overhead, especially for resource-limited Edge/Fog nodes. Thus, for hybrid computing environments, consisting of heterogeneous Edge/Fog and/or Cloud nodes, lightweight container orchestration tools are required to support both resource-limited resources at the Edge/Fog and resource-rich resources at the Cloud. Thus, in this paper, we propose a feasible approach to build a hybrid and lightweight cluster based on K3s, for the FogBus2 framework that offers containerized resource management framework. This work addresses the challenge of creating lightweight computing clusters in hybrid computing environments. It also proposes three design patterns for the deployment of the FogBus2 framework in hybrid environments, including 1) Host Network, 2) Proxy Server, and 3) Environment Variable. The performance evaluation shows that the proposed approach improves the response time of real-time IoT applications up to 29% with acceptable and low overhead., Comment: 20 pages, 10 figures

Currently, due to the advantages of light weight, simple deployment, multi-environment support, short startup time, scalability, and easy migration, container technology has been widely used in both cloud and edge/fog computing, and addresses the problem of device heterogeneity in different computing environments. On this basis, as one of the most popular container orchestration and management systems, Kubernetes almost dominates the cloud environment. However, since it is primarily designed for centralized resource management scenarios where computing resources are sufficient, the system is unstable in edge environments due to hardware limitations. Therefore, in order to realize container orchestration in the cloud and edge/fog hybrid computing environment, we propose a feasible approach to build a hybrid clustering based on K3s, which solves the problem that virtual instances in different environments cannot be connected due to IP addresses. We also propose three design patterns for deploying the FogBus2 framework into hybrid environments, including 1) Host Network Mode, 2) Proxy Server, and 3) Environment Variable.

A graph $G$ is said to satisfy the Vizing bound if $\chi(G)\leq \omega(G)+1$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. It was conjectured by Randerath in 1998 that if $G$ is a triangle-free and fork-free graph, where the fork (also known as trident) is obtained from $K_{1,4}$ by subdividing two edges, then $G$ satisfies the Vizing bound. In this paper, we confirm this conjecture., Comment: 24 pages, 1 figure

With the development of image recovery models,especially those based on adversarial and perceptual losses,the detailed texture portions of images are being recovered more naturally.However,these restored images are similar but not identical in detail texture to their reference images.With traditional image quality assessment methods,results with better subjective perceived quality often score lower in objective scoring.Assessment methods suffer from subjective and objective inconsistencies.This paper proposes a regional differential information entropy (RDIE) method for image quality assessment to address this problem.This approach allows better assessment of similar but not identical textural details and achieves good agreement with perceived quality.Neural networks are used to reshape the process of calculating information entropy,improving the speed and efficiency of the operation. Experiments conducted with this study image quality assessment dataset and the PIPAL dataset show that the proposed RDIE method yields a high degree of agreement with people average opinion scores compared to other image quality assessment metrics,proving that RDIE can better quantify the perceived quality of images., Comment: 14 pages, 8 figures, 5 tables

For any positive integer $t$, a \emph{$t$-broom} is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to $7\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and \{$t$-broom, $K_{t,t}$\}-free graphs, we improve the bound to $o(\omega^{t})$., Comment: 14 pages, 1 figure

Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles., Comment: arXiv admin note: text overlap with arXiv:2104.04898

An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the maximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|

Speech enhancement at extremely low signal-to-noise ratio (SNR) condition is a very challenging problem and rarely investigated in previous works. This paper proposes a robust speech enhancement approach (UNetGAN) based on U-Net and generative adversarial learning to deal with this problem. This approach consists of a generator network and a discriminator network, which operate directly in the time domain. The generator network adopts a U-Net like structure and employs dilated convolution in the bottleneck of it. We evaluate the performance of the UNetGAN at low SNR conditions (up to -20dB) on the public benchmark. The result demonstrates that it significantly improves the speech quality and substantially outperforms the representative deep learning models, including SEGAN, cGAN fo SE, Bidirectional LSTM using phase-sensitive spectrum approximation cost function (PSA-BLSTM) and Wave-U-Net regarding Short-Time Objective Intelligibility (STOI) and Perceptual evaluation of speech quality (PESQ)., Comment: Published in Interspeech 2019

In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for some } h\in E(\mathcal{H})\}$. A graph is \textit{outerplanar} if it can be embedded in the plane such that all its vertices lie on the outer face. A $3$-uniform hypergraph $\mathcal{H}$ is called \textit{outerplanar} if its shadow has an outerplanar embedding such that every hyperedge of $\mathcal{H}$ is the vertex set of an interior triangular face of the shadow. Cvetkovi\'c and Rowlinson conjectured in 1990 that among all outerplanar graphs on $n$ vertices, the graph $K_1+ P_{n-1}$ attains the maximum spectral radius. We show a hypergraph analogue of the Cvetkovi\'c-Rowlinson conjecture. In particular, we show that for sufficiently large $n$, the $n$-vertex outerplanar $3$-uniform hypergraph of maximum spectral radius is the unique $3$-uniform hypergraph whose shadow is $K_1 + P_{n-1}$., Comment: 13 pages, 3 figures

We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $\Delta(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar [{\em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs with positive combinatorial curvature., Comment: 10 pages, 2 figures

We study the existence of solutions of Ricci flow equations of Ollivier-Lin-Lu-Yau curvature defined on weighted graphs. Our work is motivated by\cite{NLLG} in which the discrete time Ricci flow algorithm has been applied successfully as a discrete geometric approach in detecting complex networks. Our main result is the existence and uniqueness theorem for solutions to a continuous time normalized Ricci flow. We also display possible solutions to the Ricci flow on path graph and prove the Ricci flow on finite star graph with at least three leaves converges to constant-weighted star., Comment: 20 pages

It is very challenging for speech enhancement methods to achieves robust performance under both high signal-to-noise ratio (SNR) and low SNR simultaneously. In this paper, we propose a method that integrates an SNR-based teachers-student technique and time-domain U-Net to deal with this problem. Specifically, this method consists of multiple teacher models and a student model. We first train the teacher models under multiple small-range SNRs that do not coincide with each other so that they can perform speech enhancement well within the specific SNR range. Then, we choose different teacher models to supervise the training of the student model according to the SNR of the training data. Eventually, the student model can perform speech enhancement under both high SNR and low SNR. To evaluate the proposed method, we constructed a dataset with an SNR ranging from -20dB to 20dB based on the public dataset. We experimentally analyzed the effectiveness of the SNR-based teachers-student technique and compared the proposed method with several state-of-the-art methods., Comment: Published in 2020 IEEE International Conference on Multimedia and Expo (ICME 2020)

3D moving object detection is one of the most critical tasks in dynamic scene analysis. In this paper, we propose a novel Drosophila-inspired 3D moving object detection method using Lidar sensors. According to the theory of elementary motion detector, we have developed a motion detector based on the shallow visual neural pathway of Drosophila. This detector is sensitive to the movement of objects and can well suppress background noise. Designing neural circuits with different connection modes, the approach searches for motion areas in a coarse-to-fine fashion and extracts point clouds of each motion area to form moving object proposals. An improved 3D object detection network is then used to estimate the point clouds of each proposal and efficiently generates the 3D bounding boxes and the object categories. We evaluate the proposed approach on the widely-used KITTI benchmark, and state-of-the-art performance was obtained by using the proposed approach on the task of motion detection.

In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of Tur\'an's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erd\H{o}s-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erd\H{o}s-Szekeres theorem on convex $k$-gons, as well as multiple semisaturation theorems for sequences and posets.

We verify an upper bound of Pach and T\'oth [Combinatorica 17(1997), 427-439, Discrete and Computational Geometry 36(2006), 527-552] on the midrange crossing constant. Details of their $\frac{8}{9\pi^2}$ upper bound have not been available. Our verification is different from their method and hinges on a result of Moon [J. Soc. Indust. Appl. Math. 13(1965), 506-510]. As Moon's result is optimal, we raise the question whether the midrange crossing constant is $\frac{8}{9\pi^2}$.

In this paper, we prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the Erd\H{o}s-R\'enyi random graph models $G(n,p)$ and $G(n,M)$, the random $d$-out(in)-regular directed graphs, and the space of random permutations. The main idea is using Ollivier's work on the Ricci curvature of Markov chairs on metric spaces. Here we give a cleaner form of such concentration inequality in graphs. Namely, we show that for any Lipschitz function $f$ on any graph (equipped with an ergodic random walk and thus an invariant distribution $\nu$) with Ricci curvature at least $\kappa>0$, we have \[\nu \left( |f-E_{\nu}f| \geq t \right) \leq 2\exp\left( -\frac{t^2\kappa}{7} \right).\], Comment: 22 pages

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this paper, we define a variant of Tur\'an number in hypergraphs, namely the cover Tur\'an number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general upper bound on the cover Tur\'an number of graphs and determine the cover Tur\'an density of all graphs when the uniformity of the host hypergraph equals to $3$., Comment: 14 pages

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1, BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform hypergraph $\mathcal{H}$ on $n \geq n_0$ vertices and every $2$-edge-coloring (blue and red) of $\mathcal{H}$ , there is either a blue Berge-$G_1$ or a red Berge-$G_2$ subhypergraph. We show that for every $k\geq 2$, there exists some $c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) \leq \hat{R}^{[k]}(BG_1, BG_2) \leq c_k \cdot R(G_1, G_2)^3$. Moreover, we show that for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$ such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $\hat{R}^{[k]}(BG,BG) \leq cn$., Comment: 9 pages

Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge belongs to $R$. A hypergraph $\mathcal{H}$ is called \textit{covering} if every vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we show that every covering $[3]$-uniform hypergraph on $n\geq 6$ vertices contains a Berge cycle $C_s$ for any $3\leq s\leq n$. As an application, we determine the maximum Lagrangian of $k$-uniform Berge-$C_{t}$-free hypergraphs and Berge-$P_{t}$-free hypergraphs., Comment: Title changed to "On Hamiltonian Berge cycles in $3$-uniform hypergraphs"

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $\lambda^{(p)}(H)$ of $H$ is defined as \[ \lambda^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraphs when $G$ is a path, a cycle or a star., Comment: 15 pages

We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We define a projection operator, called shadow, that maps Bayesian hypergraphs to chain graphs, and show that the Markov properties of a Bayesian hypergraph are equivalent to those of its corresponding chain graph. We extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention.

For positive integers $n$ and $e$, let $\kappa(n,e)$ be the minimum crossing number (the standard planar crossing number) taken over all graphs with $n$ vertices and at least $e$ edges. Pach, Spencer and T\'oth [Discrete and Computational Geometry 24 623--644, (2000)] showed that $\kappa(n,e) n^2/e^3$ tends to a positive constant (called midrange crossing constant) as $n\to \infty$ and $n \ll e \ll n^2$, proving a conjecture of Erd\H{o}s and Guy. In this note, we extend their proof to show that the midrange crossing constant exists for graph classes that satisfy a certain set of graph properties. As a corollary, we show that the the midrange crossing constant exists for the family of bipartite graphs. All these results have their analogues for rectilinear crossing numbers.

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs., Comment: Updated to include suggestions of the referee

The crossing number ${\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, ${\mbox {cr}}_k(G)$, is defined as the minimum of ${\mbox {cr}}(G_1)+{\mbox {cr}}(G_2)+\ldots+{\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}$ with $\cup_{i=1}^{k}G_i=G$. Pach et al. [\emph{Computational Geometry: Theory and Applications} {\bf 68} 2--6, (2018)] showed that for every $k\ge 1$, we have ${\mbox {cr}}_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right){\mbox {cr}}(G)$ and that this bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. We improve the upper bound to $\frac{1}{k^2}(1+o(1))$ as $k\rightarrow \infty$. For the class of bipartite graphs, we show that the best constant is exactly $\frac{1}{k^2}$ for every $k$. The results extend to the rectilinear variant of the $k$-planar crossing number., Comment: 13 pages, 1 figure, 1 table

In the cops and robber games played on a simple graph $G$, Aigner and Fromme's lemma states that one cop can guard a shortest path in the sense that the robber cannot enter this path without getting caught after finitely many steps. In this paper, we extend Aigner and Fromme's lemma to cover a larger family of graphs and give metric characterizations of these graphs. In particular, we show that a generalization of block graphs, namely vertebrate graphs, are 1-guardable. We use this result to give the cop number of some special class of multi-layer generalized Peterson graphs., Comment: fixing typos

We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$., Comment: 8 pages, 2 figures

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is $n$ while its local dimension is only $3$. Hiraguchi (1955) proved that the maximum dimension of a poset of order $n$ is $n/2$. However, we find a very different result for local dimension, proving a bound of $\Theta(n/\log n)$. This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the $n$-dimensional Boolean lattice is $\Omega(n/\log n)$ and make progress toward resolving a version of the removable pair conjecture for local dimension., Comment: 13 pages, 1 figure

An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$., Comment: 17 pages, fixed an error in the proof of Theorem 3 using Matroid methods

For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest integer $m$ such that there exists a $k$-uniform hypergraph $\mathcal{H}$ on $m$ edges such that any coloring of the edges of $\mathcal{H}$ with $r$ colors yields a monochromatic copy of $\mathcal{G}$. Let $\mathcal{P}_{n,k-1}^{(k)}$ denote the $k$-uniform tight path on $n$ vertices. Dudek, Fleur, Mubayi and R\H{o}dl showed that the size-Ramsey number of tight paths $\hat R(\mathcal{P}_{n,k-1}^{(k)}, 2) = O(n^{k-1-\alpha} (\log n)^{1+\alpha})$ where $\alpha = \frac{k-2}{\binom{k-1}{2}+1}$. In this paper, we improve their bound by showing that $\hat R(\mathcal{P}_{n,k-1}^{(k)}, r) = O(r^k (n\log n)^{k/2})$ for all $k\geq 3$ and $r\geq 2$., Comment: 9 pages

We give an explicit extension of Spencer's result on the biplanar crossing number of the Erdos-Renyi random graph $G(n,p)$. In particular, we show that the k-planar crossing number of $G(n,p)$ is almost surely $\Omega((n^2p)^2)$. Along the same lines, we prove that for any fixed $k$, the $k$-planar crossing number of various models of random $d$-regular graphs is $\Omega ((dn)^2)$ for $d > c_0$ for some constant $c_0=c_0(k)$.

In 1929, Hermann Weyl derived the massless solutions from the Dirac equation - the relativistic wave equation for electrons. Neutrinos were thought, for decades, to be Weyl fermions until the discovery of the neutrino mass. Moreover, it has been suggested that low energy excitations in condensed matter can be the solutions to the Weyl Hamiltonian. Recently, photons have also been proposed to emerge as Weyl particles inside photonic crystals. In all cases, two linear dispersion bands in the three-dimensional (3D) momentum space intersect at a single degenerate point - the Weyl point. Remarkably, these Weyl points are monopoles of Berry flux with topological charges defined by the Chern numbers. These topological invariants enable materials containing Weyl points to exhibit a wide variety of novel phenomena including surface Fermi arcs, chiral anomaly, negative magnetoresistance, nonlocal transport, quantum anomalous Hall effect, unconventional superconductivity[15] and others [16, 17]. Nevertheless, Weyl points are yet to be experimentally observed in nature. In this work, we report on precisely such an observation in an inversion-breaking 3D double-gyroid photonic crystal without breaking time-reversal symmetry., Comment: 4 pages, 3 figures

We demonstrate how broadband angular selectivity can be achieved with stacks of one-dimensionally periodic photonic crystals, each consisting of alternating isotropic layers and effective anisotropic layers, where each effective anisotropic layer is constructed from a multilayered metamaterial. We show that by simply changing the structure of the metamaterials, the selective angle can be tuned to a broad range of angles; and, by increasing the number of stacks, the angular transmission window can be made as narrow as desired. As a proof of principle, we realize the idea experimentally in the microwave regime. The angular selectivity and tunability we report here can have various applications such as in directional control of electromagnetic emitters and detectors., Comment: 5 pages, 5 figures

The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of {\Phi}(E,L), where E is the particle energy and L the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem, the galaxy model {\Phi}(E,L) is stable if the distribution {\Phi} is monotonically decreasing with respect to the particle energy E. On the other hand, the stability of {\Phi}(E,L) remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin, we constuct examples of unstable galaxy models of f(E,L) and f(E) in which f fails to be monotone in E.

We show that a metallic plate with fractal-shaped slits can be homogenitized as a plasmonic metamaterial with plasmon frequency dictated by the fractal geometry. Owing to the all-dimensional subwavelength nature of the fractal pattern, our system supports both transverse-electric and transverse-magnetic surface plasmons. As a result, this structure can be employed to focus light sources with all-dimensional subwavelength resolutions and enhanced field strengths. Microwave experiments reveal that the best achievable resolution is only, and simulations demonstrate that similar effects can be realized at infrared frequencies with appropriate designs., Comment: 13 pages, 4 figures