AI 中文总结
该研究提出带局部约束的组合结构构造统一框架，拓展配置模型，通过2-均匀性刻画贪心采样渐近均匀性条件，给出渐近分布与计数公式，明确临界指数，还可处理禁止边及边着色图采样问题。
AI 中文摘要
我们提出了一个在局部约束下构造组合结构的统一框架。该方法拓展了用于给定度序列随机图的configuration model（配置模型），涵盖多种特例，包括二分图、有向图、定向图、边着色（二分）图以及（有向）超图。\n通过将半边匹配重新表述为辅助图中的独立集问题，我们识别出2-均匀性——这一性质刻画了贪心采样何时能保持渐近均匀性。我们对所有2-均匀图进行了分类，证明仅有两类（配置空间与二分配置空间）具有无界独立数，从而支持渐近情形。我们的主定理给出了构型的渐近采样分布与计数公式，当边数m趋于无穷且最大度d_max=O(m^{1/4}/log m)时，误差项阶为O(d_max^4 log m/m + d_max^2 (log m)^2/m)。这解决了长期存在的O(m^{1/4-τ})界（τ为某固定正数）问题，明确了临界指数。\n此外，只要每个顶点参与的禁止边数不超过O(m^{1/4}/log m)，我们的定理就能纳入禁止边情形。特别地，这使得通过逐次构造各颜色子图，来采样每个颜色类具有给定度序列的边着色图成为可能。
英文摘要
We develop a unified framework for constructing combinatorial structures under local constraints. Our approach extends the configuration model for random graphs with a prescribed degree sequence, and covers many special cases, including bipartite graphs, directed graphs, oriented graphs, edge-colored (bipartite) graphs, and (directed) hypergraphs. By reformulating half-edge matching as an independent set problem in an auxiliary graph, we identify 2-uniformity, a property characterising when greedy sampling preserves asymptotic uniformity. We classify all 2-uniform graphs and show that only two classes, the configuration space and the bipartite configuration space, have unbounded independence number, enabling the asymptotic regime. Our main theorem then gives the asymptotic sampling distribution and enumeration formulae for configurations, with error terms of order $O(d_{\max}^4\log m/m+d_{\max}^2(\log m)^2/m)$ as the number of edges $m$ tends to infinity with maximum degree $d_{\max}=O(m^{1/4}/\log m)$. This settles the long-standing $O(m^{1/4-τ})$ bound (for some fixed $τ> 0$), making the critical exponent explicit. Furthermore, our theorem accommodates forbidden edges, provided that each vertex participates in at most $O(m^{1/4}/\log m)$ of them. In particular, this enables the sampling of edge-colored graphs with prescribed degree sequences for each color class by constructing the colored subgraphs one at a time.
