General Krivelevich–Lew–Michaeli minimum-degree conjecture

Prove that every n-vertex graph G with n>d≥1 and minimum degree satisfying δ(G)≥max{(n+d)/2−1, 2d−d(d+1)/n} is generically d-rigid.

Background

The conjecture predicts that vertex connectivity and Maxwell’s edge-count condition together suffice for generic d-rigidity. The paper establishes only restricted ranges and dense special cases, and explicitly states that the conjecture remains widely unresolved.

References

Beyond these results, \cref{con:KLM} remains wide open, even in the regular dense setting of \cref{con:KLM-regular}.

— Rigidity of complements of bounded-degree graphs  (2609.05058 - Haslegrave et al., 4 Sep 2026) in Conjecture 1, Section 1; unresolved status stated later in Section 1

The following asymptotic consequence of \cref{con:KLM} may be a more approachable first step in its high-dimensional regime. \begin{conjecture}\label{con:bounded-complement-degree} Let $D=D(n)=o(n)$. Then, uniformly over all $n$-vertex graphs $F$ with $\Delta(F)\leq D$,

\rig(\overline{F})=(1-o(1))n.

\end{conjecture}

— Rigidity of complements of bounded-degree graphs  (2609.05058 - Haslegrave et al., 4 Sep 2026) in Conjecture 3.??, Section 1, immediately following Corollary 3; label con:bounded-complement-degree