John Haslegrave

Sharp thresholds for flexible realisations in random graphs

An embedding of the vertices of a graph in the plane is flexible if the positions of vertices can be continuously deformed while maintaining the edge lengths. We say that the embedding is quasi-injective if all edge lengths are positive. Grasegger, Legerský and Schicho gave an exact combinatorial condition in terms of edge colourings for a flexible quasi-injective embedding of a given graph to exist. A simpler sufficient condition is for the graph to have an independent cutset. We show that the thresholds for both of these properties in the binomial random graph coincide with that for every vertex to be in a triangle, and in fact in the random graph process all three transitions occur simultaneously with high probability. This is joint work with Katie Clinch, Tony Huynh and Tony Nixon.