Sharp thresholds for NAC-colourings and stable cuts in random graphs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908579336814592 |
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| author | Clinch, Katie Haslegrave, John Huynh, Tony Nixon, Anthony |
| author_facet | Clinch, Katie Haslegrave, John Huynh, Tony Nixon, Anthony |
| contents | NAC-colourings of graphs correspond to flexible quasi-injective realisations in $\mathbb {R} ^2$. A special class of NAC-colourings are those that arise from stable cuts. We give sharp thresholds for the random graph to have no stable cut and to have no NAC-colouring via exact hitting-time results: with high probability, the random graph process gains both properties at the precise time that every vertex is in a triangle. Our thresholds complement recent results on the thresholds for the random graph to be generically or globally rigid in $\mathbb {R} ^d$, and for all injective realisations to be globally rigid in $\mathbb {R} $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp thresholds for NAC-colourings and stable cuts in random graphs Clinch, Katie Haslegrave, John Huynh, Tony Nixon, Anthony Combinatorics Probability 05C80 (primary) 52C25, 05C15 (secondary) NAC-colourings of graphs correspond to flexible quasi-injective realisations in $\mathbb {R} ^2$. A special class of NAC-colourings are those that arise from stable cuts. We give sharp thresholds for the random graph to have no stable cut and to have no NAC-colouring via exact hitting-time results: with high probability, the random graph process gains both properties at the precise time that every vertex is in a triangle. Our thresholds complement recent results on the thresholds for the random graph to be generically or globally rigid in $\mathbb {R} ^d$, and for all injective realisations to be globally rigid in $\mathbb {R} $. |
| title | Sharp thresholds for NAC-colourings and stable cuts in random graphs |
| topic | Combinatorics Probability 05C80 (primary) 52C25, 05C15 (secondary) |
| url | https://arxiv.org/abs/2510.05838 |