The edge-redundant rigidity conjecture for complete graphs in d\ell_\infty^d

Let d2d\geq 2, let n2d+2n\geq 2d+2, and let KnK_n denote the complete graph on nn vertices. A graph is edge-redundantly rigid in a normed space if it has a framework that is infinitesimally rigid and remains infinitesimally rigid after deleting any edge. Edge-redundant rigidity conjecture. For every d2d\geq 2 and every n2d+2n\geq 2d+2, the complete graph KnK_n is edge-redundantly rigid in d\ell_\infty^d.

The preceding proposition rules out K2d+1K_{2d+1}, so the conjecture asserts that the stated threshold is sufficient as well as necessary. The supplied excerpt gives no evidence that the conjecture has been resolved.

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Primary source

James Cruickshank, Sean Dewar and Derek Kitson, “Algebraic connectivity in normed spaces”, arXiv:2508.00134 (2025).

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