The chromatic number of random intersection graphs

Abstract

In the random intersection graph model G(n,m,P(m)) to each vertex from a vertex set V (|V| = n) we assign, independently from all other vertices, a random set of its features W (v) from an auxiliary set W (|W| = m = m(n)). The cardinality of W (v) is chosen according to a given probability distribution P(m) and W (v) is chosen uniformly at random from all subsets of W of this cardinality. We connect vertices v and u by an edge if the sets W (v) and W (u) intersect. We present some recent results concerning colouring random intersection graphs. It was shown in [1] that for some range of parameters with high probability 1 the chromatic number χ(G(n,m,P(m))) is asymptotically equal to the clique number ω(G(n,m,P(m))), for P(m) binomial Bin(m, p). We reprove this result in a very simple way and we partially resolve the problem for a wider range of parameters. Our results are related to the problem of determining the chromatic index of the random hypergraph H = {W, E}, where E = {W (v), v ∈ V}, i.e. they are linked to the problem of generalising Vizing’s theorem for hypergraphs.