5-regular graphs and the 3-dimensional rigidity matroid
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913915410055168 |
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| author | Monks, Rebecca Nixon, Anthony |
| author_facet | Monks, Rebecca Nixon, Anthony |
| contents | A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity matroid of $G$. The combinatorial nature of this matroid is well understood when $d=1,2$ but open when $d\geq 3$. Jackson and Jordán 2005 characterised independence in this matroid for connected graphs with minimum degree at most $d+1$ and maximum degree at most $d+2$. Their characterisation is known to be false for $(d+2)$-regular graphs when $d\geq 4$ but when $d=3$ it remained open. Indeed they conjectured that their characterisation extends to 5-regular graphs when $d=3$. The purpose of this article is to prove their conjecture. That is, we prove that every 5-regular graph that has at most $3n-6$ edges in any subgraph on $n\geq 3$ vertices is independent in the generic 3-dimensional rigidity matroid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22214 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 5-regular graphs and the 3-dimensional rigidity matroid Monks, Rebecca Nixon, Anthony Combinatorics 52C25, 05C10 A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity matroid of $G$. The combinatorial nature of this matroid is well understood when $d=1,2$ but open when $d\geq 3$. Jackson and Jordán 2005 characterised independence in this matroid for connected graphs with minimum degree at most $d+1$ and maximum degree at most $d+2$. Their characterisation is known to be false for $(d+2)$-regular graphs when $d\geq 4$ but when $d=3$ it remained open. Indeed they conjectured that their characterisation extends to 5-regular graphs when $d=3$. The purpose of this article is to prove their conjecture. That is, we prove that every 5-regular graph that has at most $3n-6$ edges in any subgraph on $n\geq 3$ vertices is independent in the generic 3-dimensional rigidity matroid. |
| title | 5-regular graphs and the 3-dimensional rigidity matroid |
| topic | Combinatorics 52C25, 05C10 |
| url | https://arxiv.org/abs/2506.22214 |