AI 中文总结
本文证明补图最大度不超2且无三角形或正方形分量的图在允许的最大维度中刚性，确定K₂ₘ删完美匹配图的刚性最大维度，解决Lew猜想并改进有界度图补图的刚性界。
AI 中文摘要
Maxwell 观察到，$\boldsymbol{\text{R}}^d$ 上任意 $n$ 个顶点的刚性通用框架图至少有 $dn-\binom{d+1}{2}$ 条边。本文证明，补图的最大度不超过 2 且没有分量同构于三角形或正方形的图，在该观察允许的最大维度中是刚性的。特别地，这确定了从完全图 $K_{2m}$ 中删除一个完美匹配得到的图是刚性的精确最大维度，解决了 Lew 近期提出的猜想。我们还推导了更一般的有界度图补图的刚性界，其显著改进了现有的基于度的界。
英文摘要
Maxwell observed that the graph of any rigid generic framework in $\mathbb{R}^d$ on $n$ vertices has at least $dn-\binom{d+1}{2}$ edges. In this article we prove that graphs whose complement has maximum degree at most two and no component isomorphic to a triangle or a square are rigid in the maximum dimension allowed by this observation. In particular, this determines the precise maximum dimension in which the graph obtained from a complete graph $K_{2m}$ by deleting a perfect matching is rigid, resolving a recent conjecture of Lew. We also deduce bounds on the rigidity of complements of bounded-degree graphs more generally, which significantly improve existing degree-based bounds.
Comments14 pages, 1 figure
