发表机构
Oregon State University（俄勒冈州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Watkins猜想对无限群成立，确定了无限群的Cayley指数，并构造了大量具有特定性质的Cayley图，提出了恢复序数层级的新工具。
AI 中文摘要
我们证明，在每一个无限基数下，每个既不是指数大于2的交换群也不是广义双循环群的群都承认一个图正则表示，从而解决了Watkins猜想的无限群部分。我们还确定了每个无限群的Cayley指数：根据其代数类型，该指数为$1$、$2$或$8$，并且在每种情况下该指数都由一个连通的Cayley图达到。对于每个基数为$\kappa$的无限群$G$，我们构造了$2^\kappa$个两两不同构的Cayley图，这些图恰好具有不可避免的反演对对称性，直径为2，并且每一对不同顶点都有$\kappa$个公共邻点。主要工具从具有有界度误差的交替邻接基线中恢复一个连续的序数层级：稳健的有限模式识别初始类，连续的孪生商恢复各层，它们的有限例外包确定平移作用。该重构不依赖于群作用，并且在额外的逐层有界度编辑下是稳定的。进一步的结果给出了正则情形下具有指定有限数据的闭Cantor立方体族、尖锐的依赖于共尾性的图性质，以及最优的三值最短路径度量。
英文摘要
We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group $G$ of cardinality $κ$, we construct $2^κ$ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and $κ$ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.
Comments36 pages, 5 figures, 2 appendices, 34 references. v3: proof overview rewritten; definitions, notation, and figures clarified; baseline-recovery argument clarified; inverse-orientation step in Appendix A corrected and expanded; complexity remark justified; countable-case attribution clarified. Main theorem statements unchanged
