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arXiv · 2510.05838

Sharp thresholds for NAC-colourings and stable cuts in random graphs

Katie Clinch·John Haslegrave·Tony Huynh·Anthony Nixon
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Abstract

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} $.

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Katie Clinch, John Haslegrave, Tony Huynh, Anthony Nixon. 2025-10-07. Sharp thresholds for NAC-colourings and stable cuts in random graphs. https://arxiv.org/abs/2510.05838

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