组合数学：2025–2026 年度综述

An annual survey of the math.CO arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 100 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Combinatorial and Random Matrix Theory (26.0%) — Studies random matrices and the combinatorial probabilistic structures arising in matrix models.
• Bounds and Conjectures in Combinatorial and Spectral Graph Theory (12.8%) — Proves bounds and explores conjectures using spectral and combinatorial techniques on graphs.
• Symmetric Polynomials and Partitions in Combinatorial Identities (11.2%) — Investigates symmetric polynomials, integer partitions, and the combinatorial identities connecting them.
• Graph Theory: Structure, Energy, and Extremal Properties (9.3%) — Studies graph structure, graph energy, and the extremal properties governing the maximum size or density of graphs.
• Finite Groups, Cayley Graphs, and Automorphisms (7.2%) — Analyses finite groups, their Cayley graphs, and automorphism groups.
• Graph Coloring, Chromatic Numbers, and Ramsey Theory (6.9%) — Examines graph colourings, chromatic numbers, and Ramsey-type results on unavoidable substructures.
• Combinatorial Set Theory and Algebraic Structures (6.4%) — Explores combinatorial set theory and the algebraic structures underlying set systems.
• Intersecting Families and Combinatorial Structures (5.5%) — Analyses intersecting families of sets and related extremal combinatorial configurations.
• Phylogenetic Trees, Spanning Trees, and Network Structures (4.9%) — Studies phylogenetic trees, spanning trees, and the combinatorial structure of networks.
• Degree Constraints and Minimum Degree Conjectures in Graph Theory (4.7%) — Explores degree constraints and conjectures relating minimum degree to other graph properties.
• Rank Metric Codes Bounds and Geometric Properties (3.0%) — Studies rank-metric codes, their bounds, and the geometric properties of their constructions.
• Cycle Lengths and Structural Properties in Graph Theory (2.2%) — Investigates cycle lengths and the structural properties they impose on graphs.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.