On the independence number and Hamiltonicity of uniform random intersection graphs

In the uniform random intersection graphs model, denoted by Gn,m,λ, to each vertex v we assign exactly λ randomly chosen labels of some label set M of m labels and we connect every pair of vertices that has at least one label in common. In this model, we estimate the independence number α(Gn,m,λ), for the wide range m=⌊nα⌋,α<1 and λ=O(m1/4). We also prove the Hamiltonicity of this model by an interesting combinatorial construction. Finally, we give a brief note concerning the independence number of Gn,m,p random intersection graphs, in which each vertex chooses labels with probability p.