Random Intersection Graphs, Gn,m,p, is a class of random graphs introduced in Karoski (1999)  where each of the n vertices chooses independently a random subset of a universal set of m elements. Each element of the universal sets is chosen independently by some vertex with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that m=nα⌉, for any real α different than one, we establish here, for the first time, a sharp threshold for the graph property "Contains a Hamilton cycle". Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection graph model. © 2010 Elsevier B.V. All rights reserved.
Efthymiou, C., & Spirakis, P. G. (2010). Sharp thresholds for Hamiltonicity in random intersection graphs. Theoretical Computer Science, 411(40–42), 3714–3730. https://doi.org/10.1016/j.tcs.2010.06.022