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arXiv · 2609.05058

Rigidity of complements of bounded-degree graphs

John Haslegrave·Peleg Michaeli·Anthony Nixon
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Abstract

Maxwell observed that the graph of any rigid generic framework in $\mathbb{R}^d$ on $n$ vertices has at least $dn-\binom{d+1}{2}$ edges. In this article we prove that graphs whose complement has maximum degree at most two and no component isomorphic to a triangle or a square are rigid in the maximum dimension allowed by this observation. In particular, this determines the precise maximum dimension in which the graph obtained from a complete graph $K_{2m}$ by deleting a perfect matching is rigid, resolving a recent conjecture of Lew. We also deduce bounds on the rigidity of complements of bounded-degree graphs more generally, which significantly improve existing degree-based bounds.

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John Haslegrave, Peleg Michaeli, Anthony Nixon. 2026-09-04. Rigidity of complements of bounded-degree graphs. https://arxiv.org/abs/2609.05058

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