Treewidth of Erdös-Rényi random graphs, random intersection graphs, and scale-free random graphs

We prove that the treewidth of an Erdös-Rényi random graph G(n, m) is, with high probability, greater than βn for some constant β> 0 if the edge/vertex ratio m n is greater than 1.073. Our lower bound m n> 1.073 improves the only previously-known lower bound established in [19]. We also study the treewidth of random graphs under two other random models for large-scale complex networks. In particular, our result on the treewidth of random intersection graphs strengths a previous observation in [16] on the average case behavior of the gate matrix layout problem. For scale-free random graphs based on the Barabási-Albert preferential-attachment model, our result shows that if more than 12 vertices are attached to a new vertex, then the treewidth of the obtained network is linear in the size of the network with high probability