Combinatorial sufficient conditions for graph rigidity and applications to random graphs

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 Jord\'an, Liu, and Vill\'anyi (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.