With the widespread application of multi-view data across various domains, multi-view unsupervised feature selection (MUFS) has achieved remarkable progress in both feature selection (FS) and missing-view completion. However, existing MUFS methods typically rely on centralized servers, which not only fail to meet privacy requirements in distributed settings but also suffer from suboptimal FS quality and poor convergence. To overcome these challenges, we propose a novel federated incomplete MUFS method (Fed-IMUFS), which integrates a fractional Sparsity-Guided Whale Optimization Algorithm (SGWOA) and Tensor Alternating Learning (TAL). Within this federated learning framework, each client performs local optimization in two stages: in the first stage, SGWOA introduces an $L_{2,1}$ proximal projection to enforce row-sparsity in the FS weight matrix, while fractional-order dynamics and fractal-inspired elite kernel injection mechanisms enhance global search ability, yielding a discriminative and stable weight matrix; in the second stage, based on the obtained weight matrix, an alternating optimization framework with tensor decomposition is employed to iteratively complete missing views while simultaneously optimizing low-dimensional representations to preserve cross-view consistency, with the objective function gradually minimized until convergence. During federated training, the server employs an aggregation and distribution strategy driven by normalized mutual information, where clients upload only their local weight matrices and quality indicators, and the server adaptively fuses them into a global FS matrix before distributing it back to clients. This process achieves consistent FS across clients while safeguarding data privacy. Comprehensive evaluations on CEC2022 and several incomplete multi-view datasets confirm that Fed-IMUFS outperforms state-of-the-art methods, delivering stronger global optimization capability, higher-quality feature selection, faster convergence, and more effective handling of missing views.

The Borodin-Kostochka conjecture states that for any graph $ G $ with $ \Delta(G) \geq 9 $, we have $ \chi(G) \leq \max\{\Delta(G)-1, \omega(G)\} $. In this paper, we study the structure of potential counterexamples by partitioning vertices according to the number of neighbors they have in a fixed maximum clique. This approach provides a sufficient condition for $ \chi(G) \leq \Delta(G)-1 $. Consequently, we confirm the conjecture for any $ \overline{K_{1, t}} $-free graph $ G $ with $ t \geq 3 $ and $ \Delta(G) \geq 2t+1 $, strengthening and extending the recent work of Lan and Lin in 2024.