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，地区常被划分为选区以选举代表。选区的划分方式可能影响选举结果，而通过操纵选区划分使特定群体获益的行为被称为"杰利蝾螈"。检测杰利蝾螈可能异常困难，但一种算法方法是将现行选区方案与大量随机抽样方案进行比较，以判断其是否为异常值。重组马尔可夫链常用于此类随机抽样：随机选取两个选区，合并后再以新方式拆分。该方法在实践中效果良好，但其理论基础仍不完善。例如，尚不清楚重组马尔可夫链是否具有不可约性，即重组操作能否实现任意两个选区方案间的相互转换。重组马尔可夫链的不可约性可转化为图论问题：对于图$G$，能否通过重组操作将$G$的所有$k$连通子图（$k$个选区）划分方案构成的空间连通？我们考虑三个单连通选区，其规模分别为$k_1\pm 1$个顶点、$k_2\pm 1$个顶点和$k_3\pm 1$个顶点。我们证明，在三角晶格中任意大的三角形区域内，重组马尔可夫链具有不可约性。这是首个在严格选区规模约束下，证明非平凡重组马尔可夫链具有不可约性的结果。