In the United States, regions are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can affect who's elected, and drawing districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect gerrymandering, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used for this random sampling: randomly choose two districts, consider their union, and split this union in a new way. This works well in practice, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a graph $G$, is the space of all partitions of $G$ into $k$ connected subgraphs ($k$ districts) connected by recombination moves? We consider three simply connected districts and district sizes $k_1\pm 1$ vertices, $k_2\pm 1$ vertices, and $k3\pm 1$ vertices. We prove for arbitrarily large triangular regions in the triangular lattice, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples.

翻译：在美国，选区常被划分为若干区域以选举代表。选区的划分方式可能影响选举结果，而通过刻意划分选区使特定群体获益的行为被称为“杰利蝾螈”（gerrymandering）。尽管识别杰利蝾螈可能异常困难，但一种算法方法是将当前选区方案与大量随机采样方案进行比较，以判断其是否为离群值。重组马尔可夫链常用于此类随机采样：随机选择两个选区，考虑其并集，并以新方式分割该并集。该方法在实践中表现良好，但其理论基础仍不完善。例如，重组马尔可夫链是否具有不可约性尚不明确——即重组移动是否足以从任意选区方案转换至另一方案。重组马尔可夫链的不可约性可表述为图论问题：对于图$G$，所有将$G$划分为$k$个连通子图（$k$个选区）的空间能否通过重组移动保持连通？我们考虑三个单连通选区，其大小分别为$k_1\pm 1$个顶点、$k_2\pm 1$个顶点和$k_3\pm 1$个顶点。我们证明，对于三角格点上任意大的三角形区域，重组马尔可夫链具有不可约性。这是在严苛选区大小约束下，针对非平凡小规模或简单实例之外的重组马尔可夫链不可约性的首个证明。