K-connectivity of uniform s-intersection graphs

Abstract

Let W1,⋯,Wn be independent random subsets of [m]={1,⋯,m}. Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set {W1,⋯ Wn}, defined by the adjacency relation: Wi∼ Wj whenever |Wi∩ Wj|≥s. We show that as n,m→∞ the edge density threshold for the property that each vertex of Gs(n,m,d) has at least k neighbours is asymptotically the same as that for G s(n,m,d) being k-connected. © 2014 Elsevier B.V. All rights reserved.