Objective Many combinatorial optimization problems can be regarded as graph coloring problems. A classic topic in this field is total coloring, which combines vertex coloring and edge coloring. Previous studies and current research focus on the Total Coloring Conjecture (TCC), proposed in the 1960s. For graphs, including planar graphs, with maximum degree less than six, the correctness of the TCC has been verified through case enumeration. For planar graphs with maximum degree greater than six, the discharging technique has been used to confirm the conjecture by identifying reducible configurations and establishing detailed discharging rules. This method becomes limited when applied to planar graphs with maximum degree exactly six. Only certain restricted classes of graphs have been shown to satisfy the TCC, such as graphs without 4-cycles and graphs without adjacent triangles. More recent work demonstrates that the TCC holds for planar graphs without 4-fan subgraphs and for planar graphs with maximum average degree less than twenty-three fifths. Thus, it remains unclear whether planar graphs with maximum degree six that contain a 4-fan subgraph or have maximum average degree at least twenty-three fifths satisfy the conjecture. To address this question, this paper studies total coloring of a class of planar graphs known as nested n-pointed stars and aims to show that the TCC holds for these graphs. Methods The study relies on theoretical methods, including mathematical induction, constructive techniques, and case enumeration. An n-pointed star is obtained by connecting each edge of an n-polygon (n >= 3) to a triangle and then joining the triangle vertices not on the polygon to form a new n-polygon. Repeating this operation produces a nested n-pointed star with l layers, denoted by G(n)(l). These graphs have maximum degree exactly six. Their structural properties, including the presence of 4-fan subgraphs and maximum average degree greater than twenty-three fifths, are established. Induction on the number of layers is then used to show that G(n)(l) has a total 8-coloring: (1) G(n)(l) has a total 8-coloring; (2)G(n)(l -1) Suppose that has a total 8-coloring; (3) prove that has a G(l) (n) total 8-coloring. A graph G(n)(l) is defined as a type I graph if it has a total 7-coloring. When n= 3k , constructive arguments show that G(3k)(l) is a type I graph. The value of is considered in two cases, (k= 2m - 1)and (k = 2m) . In both cases, a total 7-coloring of G(3k)(l) is obtained by directly assigning colors to all vertices and edges. Results and Discussions Induction on the number of layers of G(n)(l) that nested n-pointed stars satisfy the Total Coloring Conjecture (Fig. 5). Five colors are assigned to the vertices and edges of to obtain a total 5-coloring (Fig. 6(a) and Fig. 8(a)). Two additional colors are then applied alternately to the edges connecting the polygons in layers 1 and 2. This produces a total 7-coloring of G(3k)(2) (Fig. 7(a) and Fig. 9(a)). After a permutation of the colors, another total 7-coloring of G(3k)(3) is obtained (Fig. 7(b) and Fig. 9(b)). The coloring pattern on the outermost layer is identical to that of G(3k)(1), which allows the same extension to construct total 7-colorings for G(3k)(4), G(3k)(5),... , G(3k)(l). Therefore, G(3k)(l) is a type I graph. Conclusions This study verifies that the Total Coloring Conjecture holds for nested n-pointed stars, which have maximum degree six and contain 4-fan subgraphs. It shows that G(3k)(l) is a type I graph. A further question arises regarding whether G(n)(l) is a type I graph when n not equal 3k. A total 7-coloring can be constructed when or n = 5, and therefore both G(4)(l) and G (l)(5) are type I graphs. For other values of n not equal 3k , whether G(n)(l) is a type I graph remains open.
Objective Low-Dose Computed Tomography (LDCT) reduces patient radiation exposure but introduces substantial noise and artifacts into reconstructed images. Convolutional Neural Network (CNN)-based denoising approaches are limited by local receptive fields, which restrict their abilities to capture long-range dependencies. Transformer-based methods alleviate this limitation but incur quadratic computational complexity relative to image size. In contrast, State Space Model (SSM)-based Mamba frameworks achieve linear complexity for long-range interactions. However, existing Mamba-based methods often suffer from information loss and insufficient noise suppression. To address these limitations, we propose the Wave-MambaCT model. Methods The proposed Wave-MambaCT model adopts a multi-scale framework that integrates Discrete Wavelet Transform (DWT) with a Mamba module based on the SSM. First, DWT performs a two-level decomposition of the LDCT image, decoupling noise from Low-Frequency (LF) content. This design directs denoising primarily toward the High-Frequency (HF) components, facilitating noise suppression while preserving structural information. Second, a residual module combined with a Spatial-Channel Mamba (SCM) module extracts both local and global features from LF and HF bands at different scales. The noise-free LF features are then used to correct and enhance the corresponding HF features through an attention-based Cross-Frequency Mamba (CFM) module. Finally, inverse wavelet transform is applied in stages to progressively reconstruct the image. To further improve denoising performance and network stability, multiple loss functions are employed, including L1 loss, wavelet-domain LF loss, and adversarial loss for HF components. Results and Discussions Extensive experiments on the simulated Mayo Clinic datasets, the real Piglet datasets, and the hospital clinical dataset DeepLesion show that Wave-MambaCT provides superior denoising performance and generalization. On the Mayo dataset, a PSNR of 31.652 8 dB is achieved, which is higher than that of the suboptimal method DenoMamba (PSNR 31.421 9 dB), while MSE is reduced to 0.000 74 and SSIM and VIF are improved to 0.885 1 and 0.462 9, respectively (Table 1). Visual results (Figs. 4 similar to 6) demonstrate that edges and fine details such as abdominal textures and lesion contours are preserved, with minimal blurring or residual artifacts compared with competing methods. Computational efficiency analysis (Table 2) indicates that Wave-MambaCT maintains low FLOPs (17.213 5 G) and parameters (5.391 3 M). FLOPs are lower than those of all networks except RED-CNN, and the parameter count is higher only than those of RED-CNN and CTformer. During training, 4.12 minutes per epoch are required, longer only than RED-CNN. During testing, 0.146 3 seconds are required per image, which is at a medium level among the compared methods. Generalization tests on the Piglet datasets (Figs. 7, 8, Tables 3, 4) and DeepLesion (Fig. 9) further confirm the robustness and generalization capacity of Wave-MambaCT.In the proposed design, HF sub-bands are grouped, and noise-free LF information is used to correct and guide their recovery. This strategy is based on two considerations. First, it reduces network complexity and parameter count. Second, although the sub-bands correspond to HF information in different orientations, they are correlated and complementary as components of the same image. Joint processing enhances the representation of HF content, whereas processing them separately would require a multi-branch architecture, inevitably increasing complexity and parameters. Future work will explore approaches to reduce complexity and parameters when processing HF sub-bands individually, while strengthening their correlations to improve recovery. For structural simplicity, SCM is applied to both HF and LF feature extraction. However, redundancy exists when extracting LF features, and future studies will explore the use of different Mamba modules for HF and LF features to further optimize computational efficiency. Conclusions Wave-MambaCT integrates DWT for multi-scale decomposition, a residual module for local feature extraction, and an SCM module for efficient global dependency modeling to address the denoising challenges of LDCT images. By decoupling noise from LF content through DWT, the model enables targeted noise removal in the HF domain, facilitating effective noise suppression. The designed RSCM, composed of residual blocks and SCM modules, captures fine-grained textures and long-range interactions, enhancing the extraction of both local and global information. In parallel, the Cross-band Enhancement Module (CEM) employs noise-free LF features to refine HF components through attention-based CFM, ensuring structural consistency across scales. Ablation studies (Table 5) confirm the essential contributions of both SCM and CEM modules to maintaining high performance. Importantly, the model's staged denoising strategy achieves a favorable balance between noise reduction and structural preservation, yielding robustness to varying radiation doses and complex noise distributions.
Objective Visible and infrared images are two commonly used modalities in intelligent transportation scenarios and play a key role in vehicle re-identification.However,differences in imaging mechanisms and spectral responses lead to inconsistent visual characteristics between these modalities,which limits cross-modal vehicle re-identification.To address this problem,this paper proposes a Channel Relation Spectra Alignment(CaRS-Align)method that uses channel relation spectra,rather than channel-wise features,as the alignment target.This strategy reduces interference caused by imaging style differences at the relational-structure level.Within each modality,a channel relation spectrum is constructed to capture stable and semantically coordinated channel-to-channel relationships through correlation modeling.At the cross-modal level,the correlation between the corresponding channel relation spectra of the two modalities is maximized to achieve consistent alignment of relational structures.Experiments on the public MSVR310 and RGBN300 datasets show that CaRS-Align outperforms existing state-of-the-art methods.For example,on MSVR310,under infrared-to-visible retrieval,CaRS-Align achieves a Rank-1 accuracy of 64.35％,which is 2.58％higher than advanced existing methods. Methods CaRS-Align adopts a hierarchical optimization paradigm:(1)for each modality,a channel-channel relation spectrum is constructed by mining inter-channel dependencies,yielding a semantically coordinated relation matrix that preserves the organizational structure of semantic cues;(2)cross-modal consistency is achieved by maximizing the correlation between the relation spectra of the two modalities,enabling progressive optimization from intra-modal construction to cross-modal alignment;and(3)relation spectrum alignment is integrated with standard classification and retrieval objectives commonly used in re-identification to supervise backbone training for the vehicle re-identification model. Results and Discussions Compared with several state-of-the-art cross-modal re-identification methods on the RGBN300 and MSVR310 datasets,CaRS-Align demonstrates strong performance and achieves best or second-best results across both retrieval modes.As shown in(Table 1),on RGBN300 it attains 75.09％Rank-1 accuracy and 55.45％mean Average Precision(mAP)in the infrared-to-visible mode,and 76.60％Rank-1 accuracy and 56.12％mAP in the visible-to-infrared mode.As shown in(Table 2),similar advantages are observed on MSVR310,with 64.54％Rank-1 accuracy and 41.25％mAP in the visible-to-infrared mode,and 64.35％Rank-1 accuracy and 40.99％mAP in the infrared-to-visible mode.(Fig.4)presents Top-10 retrieval results,where CaRS-Align reduces identity mismatches in both directions(Fig.5)illustrates feature distance distributions,showing substantial overlap between intra-class and inter-class distances without CaRS-Align(Fig.5(a)),whereas clearer separation is observed with CaRS-Align(Fig.5(b)),confirming improved feature discrimination.These results indicate that modeling channel-level relational structures improves both retrieval modes,increases adaptability to modality shifts,and effectively reduces mismatches caused by cross-modal differences. Conclusions This paper proposes a visible-infrared cross-modal vehicle re-identification method based on CaRS-Align.Within each modality,a channel relation spectrum is constructed to preserve semantic co-occurrence structures.A CaRS-Align function is then designed to maximize the correlation between modalities,thereby achieving consistent alignment and improving cross-modal performance.Experiments on the MSVR310 and RGBN300 datasets demonstrate that CaRS-Align outperforms existing state-of-the-art methods in key metrics,including Rank-1 accuracy and mAP.
Objective Data-driven deep learning methods are widely applied to cancer subtyping,yet their performance depends on large training datasets with fine-grained annotations.For gigapixel Whole Slide Images(WSI),such annotations are labor-intensive and costly.Clinical data are typically stored in isolated data silos,and sharing procedures raise privacy concerns.Federated Learning(FL)enables a global model to be trained from data distributed across multiple medical centers without transmitting local data.However,in conventional FL,substantial heterogeneity across centers reduces the performance and stability of the global model. Methods A privacy-preserving FL method is proposed for gigapixel WSI in computational pathology.Weakly supervised attention-based Multiple Instance Learning(MIL)is integrated with differential privacy to support training when only slide-level labels are available.Within each client,a multi-scale attention-based MIL method is used to conduct local training on histopathology WSIs,reducing the need for costly pixel-level annotation through a weakly supervised setting.During the federated update,local differential privacy is applied to limit the risk of sensitive information leakage.Random noise drawn from a Gaussian or Laplace distribution is added to model parameters after each client's local training.Furthermore,a federated adaptive reweighting strategy is introduced to address the heterogeneity of pathological images across clients by dynamically balancing the influence of local data quantity and quality on each client's aggregation weight. Results and Discussions The proposed FL framework is evaluated on two clinical diagnostic tasks:Non-small Cell Lung Cancer(NSCLC)histologic subtyping and Breast Invasive Carcinoma(BRCA)histologic subtyping.As shown in(Table 1,Table 2,and Fig.4),the proposed FL method(Ours with DP and Ours w/o DP)achieves higher accuracy and stronger generalization than localized models and other FL approaches.Its classification performance remains competitive even when compared with the centralized model(Fig.3).These results indicate that privacy-preserving FL is a feasible and effective strategy for multicenter histopathology images and may reduce the performance degradation typically caused by data heterogeneity across centers.When the magnitude of added noise is controlled within a limited range,stable classification can also be achieved(Table 3).The two main components,the multiscale representation attention network and the federated adaptive reweighting strategy,each contribute to consistent performance improvement(Table 4).In addition,the proposed FL method maintains stable classification performance across different hyperparameter settings(Table 5,Table 6),confirming its robustness. Conclusions The proposed FL method addresses two central challenges in multicenter computational pathology:the presence of data silos and concerns over privacy.It also alleviates the performance degradation caused by inter-center data heterogeneity.As balancing model accuracy with privacy protection remains a key challenge,future work focuses on developing methods that preserve privacy while sustaining stable classification performance.
ObjectiveA novel cross-modal hashing framework driven by large language models (LLMs) is proposed to address the semantic misalignment between medical images and their corresponding textual reports. The objective is to enhance cross-modal semantic representation and improve retrieval accuracy by effectively mining and matching semantic associations between modalities.MethodsThe generative capacity of LLMs is first leveraged to produce high-quality textual descriptions of medical images. These descriptions are integrated with diagnostic reports and structured clinical data using a dual-stream semantic enhancement module, designed to reinforce inter-modality alignment and improve semantic comprehension. A structural similarity-guided hashing scheme is then developed to encode both visual and textual features into a unified Hamming space, ensuring semantic consistency and enabling efficient retrieval. To further enhance semantic alignment, a prompt-driven attention template is introduced to fuse image and text features through fine-tuned LLMs. Finally, a contrastive loss function with hard negative mining is employed to improve representation discrimination and retrieval accuracy.Results and DiscussionsExperiments are conducted on a multimodal medical dataset to compare the proposed method with existing cross-modal hashing baselines. The results indicate that the proposed method significantly outperforms baseline models in terms of precision and mean Average Precision (mAP) (Table 3; Table 4). On average, a 7.21% improvement in retrieval accuracy and a 7.72% increase in mAP are achieved across multiple data scales, confirming the effectiveness of the LLM-driven semantic mining and hashing approach.ConclusionsExperiments are conducted on a multimodal medical dataset to compare the proposed method with existing cross-modal hashing baselines. The results indicate that the proposed method significantly outperforms baseline models in terms of precision and mean Average Precision (mAP) (Table 3; Table 4). On average, a 7.21% improvement in retrieval accuracy and a 7.72% increase in mAP are achieved across multiple data scales, confirming the effectiveness of the LLM-driven semantic mining and hashing approach.
