离散数学：2025–2026 年度综述

An annual survey of the cs.DM arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

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

• Structural Graph Theory and Sparse Graph Classes (16.3%) — Focuses on structural properties, decompositions, and logical transductions of sparse, planar, and chordal graph classes.
• Tree Decompositions and Hierarchical Structures (13.4%) — Investigates tree-based decompositions, spanning trees, and hierarchical representations in combinatorial structures.
• Combinatorial Bounds and Extremal Graph Theory (12.6%) — Establishes upper and lower bounds for graph parameters such as chromatic number and covering numbers in extremal settings.
• Parameterized Complexity and Algorithmic Graph Theory (12.1%) — Analyzes the computational complexity of graph problems like domination and coloring using parameterized and fixed-parameter tractable techniques.
• Polynomial Time Algorithms and Algebraic Methods (9.6%) — Develops efficient algorithms and algebraic characterizations for graph properties, including polynomial identity testing and automata equivalence.
• Graph Class Recognition and Structural Characterizations (6.7%) — Studies recognition algorithms and structural characterizations for specific graph classes defined by forbidden minors or width parameters.
• Forbidden Subgraph Problems and Coloring Constraints (5.5%) — Examines coloring and subgraph deletion problems in graphs restricted by forbidden substructures like triangles or cycles.
• Combinatorial Optimization and Matching Problems (7.5%) — Addresses optimization challenges including maximum matching, vertex cover, and flow problems through algorithmic and approximation approaches.
• Network Dynamics and Phylogenetic Network Analysis (9.9%) — Explores structural and dynamic properties of complex networks, phylogenetic networks, and boolean networks.
• Combinatorial Conjectures and Extremal Counterexamples (6.6%) — Tests and refutes long-standing combinatorial conjectures, such as the lonely runner problem, through constructive proofs and counterexamples.

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.