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Lower Bound Theorem–type bounds for simplicial k-circuits

Show that for every simplicial k-circuit Δ on n vertices (k ≥ 2), the face numbers satisfy f_j(Δ) ≥ φ_j(k,n) for all j ≥ 0, with equality for some 1 ≤ j ≤ k if and only if Δ is a simplicial k-sphere.

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Background

Fogelsanger’s theorem implies rigidity of graphs of simplicial circuits, and the classical Lower Bound Theorem (LBT) holds for polytopal spheres and more generally pseudomanifolds. Extending LBT-type bounds from spheres/pseudomanifolds to all simplicial circuits is natural but unproved.

A solution would unify combinatorial lower bounds across a broader class central to rigidity methods.

References

As far as we are aware the following is still open.\n\nConjecture\nLet $k \geq 2$ be an integer and\n$\Delta$ be a simplicial $k$-circuit with $n$ vertices. Then $f_{j}(\Delta) \geq \phi_j(k,n)$ for all $j \geq 0$. Moreover, if equality holds for some $1 \leq j \leq k$ then $\Delta$ is a simplicial $k$-sphere.

— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach (2508.11636 - Cruickshank et al., 29 Jul 2025) in Rigidity and simplicial complexes — Some historical comments