As an emerging self-supervised graph representation learning model, graph contrastive learning has garnered significant attention. These models do not rely on manual annotations but instead leverage unsupervised contrastive learning strategies to maximize the consistency of similar graph or node representations, thereby learning high-quality graph representations. To fully exploit the structural information within graphs, we propose a graph contrastive learning model, GCL-CCSE. The model combines community detection and subgraph auxiliary models to better capture both local and global structural information during the learning process. Specifically, GCL-CCSE not only considers node-level information in the contrastive learning process but also incorporates community and subgraph-level information, avoiding comparisons solely between nodes and the entire graph, thus preserving the uniqueness of node embeddings. The model performs data augmentation through random edge dropping and feature masking, encodes using a Graph Convolutional Network (GCN), and generates positive and negative samples through community partitioning and subgraph sampling. Experiments conducted on four datasets for node classification tasks demonstrate that GCL-CCSE outperforms existing graph representation learning algorithms across multiple metrics. Additionally, ablation experiments verify the contributions of community and subgraph information to model performance, while sensitivity experiments explore the impact of subgraph size and the number of communities on model performance. This research provides new insights into graph contrastive learning and significantly enhances the performance of the model in downstream tasks.

We use the generating function approach to derive simple expressions for the factorial moments of the distance distribution over Reed-Solomon codes. We obtain better upper bounds for the error term of a counting formula given by Li and Wan, which gives nontrivial estimates on the number of polynomials over finite fields with prescribed leading coefficients and a given number of linear factors. This improvement leads to new results on the classification of deep holes of Reed-Solomon codes.

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which is used to show that such polynomials with degree $2n$ always exist provided that the number of prescribed leading coefficients is slightly less than $n/4$. Exact expressions are also obtained for fields with two or three elements and up to two prescribed leading coefficients.

We derive an asymptotic expression for the number of cubic maps on orientable surfaces when the genus is proportional to the number of vertices. Let Σ_g denote the orientable surface of genus g and θ=g/n∈ (0,1/2). Given g,n∈ ℕ with g→ ∞ and n/2-g→ ∞ as n→ ∞, the number C_{n,g} of cubic maps on Σ_g with 2n vertices satisfies C_{n,g} ∼ (g!)² α(θ) β(θ)ⁿ γ(θ)^{2g}, as g→ ∞, where α(θ),β(θ),γ(θ) are differentiable functions in (0,1/2). This also leads to the asymptotic number of triangulations (as the dual of cubic maps) with large genus. When g/n lies in a closed subinterval of (0,1/2), the asymptotic formula can be obtained using a local limit theorem. The saddle-point method is applied when g/n→ 0 or g/n→ 1/2.

Jackson and Wormald conjecture that if G is a 3‐connected n‐vertex graph with maximum degree $d\ge 4$, then G has a cycle of length $\Omega(n^{\log_{d-1}2})$. We show that this conjecture holds when $d-1$ is replaced by $\max\{64,4d+1\}$. Our proof implies a cubic algorithm for finding such a cycle.

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [ Random Structures Algorithms, 7 (1995), pp. 273--285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps.