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Rank formula in dimensions d ≥ 4 via proper {K_{d+2}, K_{d+2,d+2}}-covers

Establish that for every graph G and integer d ≥ 4, r_d(G) equals min_{F, C} {|F| + |⋃_{C∈C} C| − |C|}, where the minimum is over all F ⊆ E(G) and all proper covers C of G − F by edge-sets of subgraphs isomorphic to K_{d+2} or K_{d+2,d+2}.

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Background

In dimensions ≥4, obstructions like K_{d+2,d+2} show that 2D-style sparsity counts and 3D-style rigid-cluster covers are insufficient. The conjectured formula mirrors the C_21 analogue proven in this paper and would similarly place r_d in NP ∩ coNP.

Validating this would give a unifying cover-based rank characterization for all higher dimensions.

References

Conjecture\nFor any graph $G$\n\begin{equation}\label{eq:cof2}\nr_d(G)=\min\left{\n|F|+\mbox{$\left|\bigcup_{C\in \mathcal{C}} C\right|$}-|\mathcal{C}|\n\right}\n\end{equation}\nwhere the minimum is taken over all $F\subseteq E$ and all proper\n${K_{d+2}, K_{d+2,d+2}}$-covers $\mathcal{C}$ of $G-F$.

eq:cof2:

$r_d(G)=\min\left\{ |F|+\mbox{$\left|\bigcup_{C\in } C\right|$}-|| \right\} $

— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach (2508.11636 - Cruickshank et al., 29 Jul 2025) in Section 4.2 (Rigidity in dimensions at least four)