Degree Sum Conditions for Graph Rigidity

TL;DR

This paper establishes new degree sum conditions that guarantee the generic rigidity of graphs in Euclidean spaces, confirming conjectures and providing exact thresholds for various parameters, with applications to random graphs.

Contribution

It proves the conjecture that a linear bound with constant 1 suffices for degree sum conditions ensuring rigidity, and determines exact thresholds for specific dimensions and graph sizes.

Findings

Proved that $f(n,d) \\leq \\frac{n}{2}+d$ with $K=1$ suffices.

Established that $f(n,d)=\lceil\frac{n+d-2}{2} \rceil$ for $n\geq 29d$.

Showed that $g(n,d)\leq n+3d$ and $g(n,d)=n+d-2$ for $n\geq d(d+2)$.

Abstract

We study sufficient conditions for the generic rigidity of a graph $G$ expressed in terms of (i) its minimum degree $\delta(G)$, or (ii) the parameter $\eta(G)=\min_{uv\notin E}(\deg(u)+\deg(v))$. For each case, we seek the smallest integers $f(n,d)$ (resp.\ $g(n,d)$) such that every $n$-vertex graph $G$ with $\delta(G)\geq f(n,d)$ (resp.\ $\eta(G)\geq g(n,d)$) 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 Erd\H{o}s-R\'enyi 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.