数学最新文献

{"title":"Local multilevel preconditioned Jacobi-Davidson method for elliptic eigenvalue problems on adaptive meshes","authors":"Jianing Guo,&nbsp;Qigang Liang,&nbsp;Xuejun Xu","doi":"10.1007/s10444-026-10373-w","DOIUrl":"10.1007/s10444-026-10373-w","url":null,"abstract":"<div><p>In this work, we propose an efficient adaptive multilevel preconditioned Jacobi-Davidson (PJD) method for eigenvalue problems with singularity. Our multilevel method utilizes a local smoothing strategy to solve the preconditioned Jacobi-Davidson algebraic systems arising from adaptive finite element methods (AFEM). As a result, the algorithm holds optimal computational complexity <span>(varvec{O(N)})</span>. The theoretical analysis reveals that our method has a uniform convergence rate with respect to mesh levels and degrees of freedom. Further, the convergence rate is not affected by highly discontinuous coefficients within the domain. Numerical results verify our theoretical findings.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"52 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149189367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Robust Estimation and Inference for Time-Varying Unconditional Volatility","authors":"Adam Lee,&nbsp;Rickard Sandberg,&nbsp;Genaro Sucarrat","doi":"10.1111/jtsa.70034","DOIUrl":"https://doi.org/10.1111/jtsa.70034","url":null,"abstract":"<p>We derive a general and robust estimator of a large class of parametric specifications of time-varying unconditional volatility of financial returns, both univariate and multivariate, and establish the Consistency and Asymptotic Normality (CAN) of the estimator. A number of well-known and widely used parametric specifications, for many of which asymptotic results have not been specifically established, are contained in the class. The estimator is robust in the sense that the exact specification of the conditional volatility dynamics need not be known or estimated, and in the sense that the stochastic component need not be strictly stationary. The latter is especially important in light of recent findings, which document that financial returns are frequently characterised by a non-stationary zero-process. Our estimator is also robust to the well-known “curse of dimensionality” in multivariate models due to its equation-by-equation nature. While our estimator does not require the exact specification of the conditional volatility dynamics to be known or estimated, our results imply that the scaled GARCH(1,1) specification is well-defined under both correct and incorrect specifications. So we provide methods for its estimation in a second step. Also, due to the assumptions we rely upon, our results extend directly to the Multiplicative Error Model (MEM) interpretation of volatility models. This means our results can also be applied to other non-negative processes like volume, duration, realised volatility, dividends, unemployment, and so on. Three numerical applications illustrate the versatility of our results.</p>","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":"47 6","pages":"1139-1152"},"PeriodicalIF":1.2,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jtsa.70034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Testing and Estimation of Change Point in ARMA Model With Heavy-Tailed G-GARCH Noises","authors":"Qiang Bai,&nbsp;Shiqing Ling,&nbsp;Ke Zhu","doi":"10.1111/jtsa.70039","DOIUrl":"https://doi.org/10.1111/jtsa.70039","url":null,"abstract":"<div>\u0000 \u0000 <p>We develop a procedure for testing and estimating the change point in the autoregressive moving average (ARMA) model with heavy-tailed general generalized autoregressive conditional heteroskedasticity (G-GARCH) noises. Based on the self-weighted least absolute deviation estimator (SLADE), we propose two score-type test statistics for change point detection. Under the null hypothesis, we show that one of them converges weakly to the maxima of a Brownian bridge, and the other one converges weakly to an extreme distribution. Furthermore, we prove that the SLADE of the change point converges weakly to the location of the maxima of a double-sided random walk, and the SLADE of other parameters is asymptotically normal. Our two score-type tests and SLADE are applicable for the data with an infinite variance, while not specifying the form of G-GARCH noises. Finally, we use simulations and two real examples to demonstrate the usefulness of the proposed tests and estimator in handling the heavy-tailed data with an infinite variance.</p>\u0000 </div>","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":"47 6","pages":"1183-1197"},"PeriodicalIF":1.2,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamics and Numerical Analysis of a Multi-Patch Influenza Transmission Model with Population Migration and Daily Commuting.","authors":"Lusha Shi, Shuai Shen, Lili Liu, Xiaohong Zhang, Zhen Jin","doi":"10.1007/s11538-026-01689-1","DOIUrl":"https://doi.org/10.1007/s11538-026-01689-1","url":null,"abstract":"<p><p>Human mobility plays an important role in the disease spatial transmission. In this paper, a multi-patch SEIR Influenza model incorporating long-term population migration and daily commuting was formulated to describe Influenza interregional spreading. Mathematically, the explicit expression for basic reproduction number was derived by the next-generation matrix method, and the globally asymptotically stability of disease free equilibrium and the unique positive equilibrium were rigorously proved. Numerically, taking the end of 2025 influenza epidemic in Shanxi province as an example, the model was fitted by using influenza surveillance data, census data, and Baidu migration data from the 11 cities. A Logistic decay form of transmission rate is introduced to characterize the effects of intervention time and intervention strength. The numerical results showed a good fitting and indicate that the provincial basic reproduction number is 2.5877, while the city-level values range from 1.13-2.85. Sensitivity analysis further showed that the initial transmission intensity, infectious period, and intervention time are key factors affecting Influenza spreading, and local transmission contributes more to epidemic spread than inter-city mobility, which implies that reducing local transmission is more effective when controlling disease spreading. Moreover, under some certain conditions, early targeted control in hotspot cities can achieve effects comparable to or even better than province-wide uniform control. These findings provide theoretical support and quantitative evidence for understanding interregional influenza transmission and designing region-specific prevention and control strategies.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149164845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Submanifolds of almost quaternionic skew-Hermitian manifolds","authors":"Ioannis Chrysikos,&nbsp;Jan Gregorovič","doi":"10.1007/s10455-026-10061-4","DOIUrl":"10.1007/s10455-026-10061-4","url":null,"abstract":"<div><p>We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds <span>((M^{4n}, Q, omega ))</span>, including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"70 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Linear Prediction With Functional Observations: Why You Can Use a Simple Post-Dimension Reduction Estimator","authors":"Won-Ki Seo","doi":"10.1111/jtsa.70038","DOIUrl":"https://doi.org/10.1111/jtsa.70038","url":null,"abstract":"<div>\u0000 \u0000 <p>We study the optimal linear prediction of a random function that takes values in an infinite dimensional Hilbert space. We begin by characterizing the mean square prediction error (MSPE) associated with a linear predictor and discussing the minimal achievable MSPE. This analysis reveals that, in general, there are multiple non-unique linear predictors that minimize the MSPE, and even if a unique solution exists, consistently estimating it from finite samples is generally impossible. Nevertheless, we can define asymptotically optimal linear operators whose empirical MSPEs approach the minimal achievable level as the sample size increases. We show that, interestingly, standard post-dimension reduction estimators, which have been widely used in the literature, attain such asymptotic optimality under minimal conditions.</p>\u0000 </div>","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":"47 6","pages":"1171-1182"},"PeriodicalIF":1.2,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matching Stability for 3-Partite 3-Uniform Hypergraphs","authors":"Hongliang Lu,&nbsp;Xinxin Ma","doi":"10.1002/jgt.70089","DOIUrl":"https://doi.org/10.1002/jgt.70089","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be three integers such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-partite &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-uniform hypergraph with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; vertices in each class. Aharoni (2017) showed that if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 \u0000 &lt;mn&gt;1&lt;","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 4","pages":"650-662"},"PeriodicalIF":1.0,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Autocorrelation Functions for Point-Process Time Series","authors":"Daniel Gervini","doi":"10.1111/jtsa.70018","DOIUrl":"https://doi.org/10.1111/jtsa.70018","url":null,"abstract":"<div>\u0000 \u0000 <p>This article introduces autocorrelograms for time series of point processes. Such time series usually arise when a longer temporal or spatio-temporal point process is sliced into smaller time units; for example, when an annual process is sliced into 365 daily replications. We assume the point processes follow a doubly-stochastic Poisson model with log-Gaussian intensity functions. The proposed autocorrelograms are computationally simple and based on binning. The asymptotic distribution of the autocorrelations is established. The ability of the method to detect the patterns of common autoregressive and moving-average time series models is shown by simulation. Two examples of application to temporal and spatial point-process time series are shown, pertaining bike demand in the Divvy bike-sharing system of Chicago and street theft in Chicago, respectively.</p>\u0000 </div>","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":"47 6","pages":"1087-1097"},"PeriodicalIF":1.2,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149177504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"IS-PINN: A Fast and Efficient Training of Physics-Informed Neural Networks Using Invariant Subspace Reductions for Solving Nonlinear Diffusion Equations","authors":"Jiafu Pang,&nbsp;Shoufeng Shen,&nbsp;Guofu Yu,&nbsp;Junyang Zhang","doi":"10.1111/sapm.70324","DOIUrl":"https://doi.org/10.1111/sapm.70324","url":null,"abstract":"<div>\u0000 \u0000 <p>We propose the invariant subspace physics-informed neural network (IS-PINN), a novel approach that combines the invariant subspace method (ISM) with deep learning to solve nonlinear diffusion equations. By restricting the solution to a prescribed invariant subspace spanned by exact basis functions, IS-PINN eliminates spatial dependencies and transforms the nonlinear diffusion equation into a dynamical system. The temporal coefficients are learned by a neural network with hard-constrained initial conditions via least-squares projection. We validate IS-PINN on five problems: the semilinear heat equation, the Riabouchinsky–Proudman–Johnson equation (RPJE), the porous medium equation, the quasilinear parabolic equation, and the (1+2)D reaction–diffusion equation. Numerical results demonstrate that IS-PINN outperforms standard PINNs, achieving mean relative <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$L_2$</annotation>\u0000 </semantics></math> errors of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1.43</mn>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mn>10</mn>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$1.43times 10^{-5}$</annotation>\u0000 </semantics></math>–<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1.26</mn>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mn>10</mn>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$1.26times 10^{-4}$</annotation>\u0000 </semantics></math> over five random seeds, with mean training time reduced by approximately 74%–98%.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149205386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hamiltonian Cycles in Tough \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 \u0000 P\u0000 4\u0000 \u0000 ∪\u0000 \u0000 P\u0000 1\u0000 \u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 -Free Graphs","authors":"Songling Shan","doi":"10.1002/jgt.70090","DOIUrl":"https://doi.org/10.1002/jgt.70090","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;In 1973, Chvátal conjectured that there exists a constant &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that every &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-tough graph on at least three vertices is Hamiltonian. This conjecture has inspired extensive research and has been verified for several special classes of graphs. Notably, Jung in 1978 proved that every 1-tough &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-free graph on at least three vertices is Hamiltonian. However, the problem remains challenging even when restricted to graphs with no induced &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;∪&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the disjoint union of a path on four vertices and a one-vertex path. In 2013, Nikoghosyan conjectured that every 1-tough &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 \u0000 &lt;mo&gt;∪&lt;/mo&gt;\u0000 \u0000 &lt;msub&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 \u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"113 4","pages":"703-721"},"PeriodicalIF":1.0,"publicationDate":"2026-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149176925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}