Let C6 be the class of triangle-free graphs with no induced K-3,3 and no hole longer than six. We study whether every graph in C6 is 4-colorable. We prove that every 4-hole is contained in a unique maximal induced complete bipartite subgraph, that two such subgraphs intersect in at most one edge, and that each vertex outside one of them has at most one neighbor in it. For vertex-critical graphs, we also bound the distances between attachment fibers. A Kempe-chain argument shows that every (k-1) vertex of a triangle-free vertex-k-critical graph lies in at least induced odd holes. A complete 2 computation identifies exactly three 4-critical graphs in C6 with at most 15 vertices. Using the catalogs of triangle-free 5-critical graphs of orders 22, 23, and 24, we further prove that every triangle-free, induced-K-3,3-free graph with at most 24 vertices is 4-colorable. Hence any counterexample in C6 has at least 25 vertices. The conjecture remains open in general; the results provide structural restrictions and verification within a finite range of orders.
We developed a fully discrete scheme for the Allen-Cahn equation on polygonal meshes by combining a second-order implicit-explicit (IMEX) backward differentiation formula (BDF2) with a nonconforming virtual element method for spatial discretization. To rigorously justify the global Lipschitz condition required in the analysis, we employed an explicitly defined truncated polynomial potential that coincided with the standard quartic double-well potential on [-M, M] and had quadratic growth outside this interval. Under a suitable time-step restriction, the proposed scheme satisfied a modified discrete energy dissipation law. Optimal error estimates of order O(hk+1 + tau 2) in the L2-norm and O(hk +tau 2) in the H1-seminorm were established. In particular, a discrete Laplace operator adapted to the nonconforming virtual element space was introduced to avoid temporal order reduction in the H1-error analysis. Numerical experiments confirmed the theoretical convergence rates.
We present a robust central scheme for shallow water equations with wetting and drying that is formally second-order accurate in smooth regions. The proposed method preserves physically admissible states, particularly the non-negativity of the water height, while accurately resolving complex flows in the presence of dry areas without relying on Riemann solvers. The scheme combines a well-balanced discretization, the surface gradient formulation, with a nonlinear invariant-region-preserving limiter to ensure consistency between flux and source term discretizations, suppress spurious oscillations near wet-dry interfaces, and maintain the well-balanced property for steady states, including lake-at-rest configurations over irregular topographies. In dry and near wet-dry regions, the limiter preserves positivity, while in fully wet regions, it remains inactive so that the scheme locally recovers the original well-balanced formulation. A cutoff criterion is introduced to activate the limiter only when necessary, thereby preserving accuracy in fully wet regions. The main contribution of this work lies in the compatible coupling of the surface gradient reconstruction and invariant-region-preserving limiting within an unstaggered central framework, together with a dedicated treatment of lake-at-rest configurations involving wet-dry interfaces. Numerical experiments demonstrate second-order numerical convergence for the water height in smooth benchmark problems, as well as robustness and favorable performance compared with existing methods.
The aim of this article was to study the square-free numbers of the form [m (c)] with smooth indices, which can be seen as an extension of a result of Cao and Zhai. As an extension to the work of Cao and Zhai, the proof of Theorem 1.1 employed the division trick, which transformed an arithmetic problem into a problem concerning exponential sum estimates. The division trick proved to be more direct and concise in comparison with alternative approaches. For the sake of completeness of our results, Theorem 1.2, serving as a complement to Theorem 1.1, can be proved by invoking results from the work of Cao and Zhai. Combining Theorems 1.1 and 1.2, we obtained our full conclusion, namely, Theorem 1.3.
The first uniform bound in the normal approximation for character ratios was obtained by Fulman (2005) with the rate of convergence O(n-1/4), and later improved to O(n-1/2) by Shao and Su (2006). In this work, we use Stein's method and the Jack measure to derive a nonuniform Berry-Esseen-type bound for character ratios. Our result yields a sharper constant compared to previously known estimates. We also discuss an application of this approximation, particularly its role in random walk on the symmetric group S n.
