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

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} $.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 3.1: Kriv97
• Theorem 3.2
• Corollary 3.3
• Lemma 3.4
• proof : Proof of \ref{['thm:ERnostablecut']}