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图刚性的组合充分条件及其在随机图中的应用

Combinatorial sufficient conditions for graph rigidity and applications to random graphs

Tel Aviv University（特拉维夫大学）；Technion Israel Institute of Technology（以色列理工学院）；University of Oxford（牛津大学）

Michael Krivelevich, Alan Lew, Peleg Michaeli

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AI总结：

本文提出了图刚性的组合充分条件，并展示了随机图在不同参数下的刚性性质及应用。

摘要：

一个图G=(V,E)被称为d-刚性，如果在顶点的通用嵌入中，任何保持边长的连续运动都会保持所有非相邻顶点对之间的距离。在本文中，我们提出了关于随机图刚性的几个新结果。特别是，我们证明存在c>0，使得对于p≥2 log n/n，二项式随机图G(n,p)以高概率（whp）是⌊c n p⌋-刚性。这在常数c方面是尖锐的，补充了Peled和Peleg（在p= o(n^{-1/2})的范围内）以及Jordán、Liu和Villányi（在常数p范围内）最近的结果。此外，我们证明对于每一个固定的d≥2和r≥501d，一个随机r-正则图以高概率d-刚性，且对于100/n≤p≤2 log n/n，二项式随机图G(n,p)包含以高概率一个⌊np/251⌋-刚性的子图，其顶点数至少为(1-e^{-np/2})n。这两个结果在乘法常数方面是尖锐的。此外，我们还提出了一种新的刚性充分条件，以图的最小码度（图中一对顶点的共同邻居数的最小值）来表示。我们论证中的主要工具是一个新的组合充分条件，它为Whiteley的顶点分裂引理和Crapo、Lindemann、Lew、Nevo、Peled和Raz等人发展的“刚性划分”方法提供了共同的推广。

英文摘要：

A graph $G=(V,E)$ is called $d$-rigid if, for a generic embedding of its vertices in $\mathbb{R}^d$, every edge-length preserving continuous motion of the vertices preserves the distances between all pairs of non-adjacent vertices as well. In this paper, we present several new results on the rigidity of random graphs. In particular, we show that there exists $c>0$ such that, for $p\ge 2 \log{n}/n$, the binomial random graph $G(n,p)$ is with high probability (whp) $\lfloor c n p\rfloor$-rigid. This is sharp up to the constant $c$, and complements recent results of Peled and Peleg (in the regime $p= o(n^{-1/2})$), and of Jordán, Liu, and Villányi (in the constant $p$ regime). Moreover, we show that for every fixed $d\ge 2$ and $r\ge 501d$, a random $r$-regular graph is whp $d$-rigid, and that for $100/n\le p\le 2\log{n}/n$, the binomial random graph $G(n,p)$ contains whp an $\lfloor np/251\rfloor$-rigid subgraph with at least $(1-e^{-np/2})n$ vertices. Both results are sharp up to the multiplicative constant. In addition, we present a new sufficient condition for rigidity in terms of the minimum codegree of the graph (the minimum number of common neighbours of a pair of vertices in the graph). A main tool in our arguments is a new combinatorial sufficient condition for rigidity, which provides a common generalization to Whiteley's vertex-splitting lemmas, and to the "rigid partitions" method, developed in works by Crapo, Lindemann, Lew, Nevo, Peled and Raz, and by the present authors.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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