Degree Sum Conditions for Graph Rigidity
Tibor Jordán, Xuemei Liu, Soma Villányi
TL;DR
The paper resolves when graphs are generically rigid in ${ m R}^d$ from two concrete nonlocal parameters: minimum degree and degree-sum of nonedges. It proves the conjecture of Krivelevich–Lew–Michaeli with a sharp linear bound in $d$, showing $oxed{f(n,d)\le rac{n}{2}+d-1}$ for all $n,d$ and, for $n box{≥}29d$, equality $oxed{f(n,d)=ig ceil rac{n+d-2}{2}ig ceil}$; it also establishes $g(n,d)\le n+3d$, with exact $g(n,d)=n+d-2$ when $n\ge d(d+2)$. For low dimensions, the authors determine exact values for $d=2,3$, confirming the conjectures there, and they apply these results to random graphs, proving that $G(n,1/2)$ is a.a.s. $d$-rigid for $d(n) o rac{7}{32}n$ (up to lower-order terms). Their approach hinges on a refined rigidity matroid framework, cone and extension arguments, and rank-contribution techniques that also yield global rigidity consequences via dimension-dropping results.
Abstract
We study sufficient conditions for the generic rigidity of a graph $G$ expressed in terms of (i) its minimum degree $δ(G)$, or (ii) the parameter $η(G)=\min_{uv\notin E}(°(u)+°(v))$. For each case, we seek the smallest integers $f(n,d)$ (resp.\ $g(n,d)$) such that every $n$-vertex graph $G$ with $δ(G)\geq f(n,d)$ (resp.\ $η(G)\geq g(n,d)$) is rigid in $\mathbb{R}^d$. Krivelevich, Lew, and Michaeli conjectured that there is a constant $K>0$ such that $f(n,d)\leq \frac{n}{2}+Kd$ for all pairs $n,d$. We give an affirmative answer to this conjecture by proving that $K=1$ suffices. For $n\geq 29d$, we obtain the exact result $f(n,d)=\lceil\frac{n+d-2}{2} \rceil$. Next, we prove that $g(n,d)\leq n+3d$ for all pairs $n,d$, and establish $g(n,d)=n+d-2$ when $n\geq d(d+2)$. For $d=2,3,$ we determine the exact values of $f(n,d)$ and $g(n,d)$ for all $n$, confirming another conjecture of Krivelevich, Lew, and Michaeli in these low-dimensional special cases. As an application, we prove that the Erdős-Rényi random graph $G(n,1/2)$ is a.a.s.\ rigid in $\mathbb{R}^d$ for $d=d(n)\sim \frac{7}{32} n$. This result provides the first linear lower bound for $d(n)$, and it answers a question of Peled and Peleg.