A NOTE ON HAMILTONICITY OF UNIFORM RANDOM INTERSECTION GRAPHS

We consider a collection of n independent random subsets of [m] = {1, 2, ... , m} that are uniformly distributed in the class of subsets of size d, and call any two subsets adjacent whenever they intersect. This adjacency relation defines a graph called the uniform random intersection graph and denoted by G(n,m,d). We fix d = 2,3, ... and study when, as n, m -> infinity, the graph G(n,m,d) contains a Hamilton cycle (the event denoted G(n,m,d) is an element of H). We show that P(G(n,m,d) is an element of H) = o(1) for d(2)nm(-1) - ln m - 2 ln ln m -> -infinity and P(G(n,m,d) is an element of H) = 1 - o(1) for 2nm(-1) - ln m - ln ln m -> +infinity.