Random Subgraphs of Cayley Graphs overp-Groups

Abstract

The subject of this paper is the size of the largest component in random subgraphs of Cayley graphs, Xn, taken over a class of p -groups, Gn. Gnconsists of p -groups, Gn, with the following properties: (i)Gn/Φ(Gn) ∼ =Fpn, where Φ(Gn) is the Frattini subgroup and (ii) | Gn| ≤nKn, where K is some positive constant. We consider Cayley graphs Xn=Γ(Gn, Sn′), where Sn′=Sn∪Sn−1, and Snis a minimal Gn-generating set. By selecting Gn-elements with the independent probability λnwe induce random subgraphs of Xn. Our main result is, that there exists a positive constant c> 0 such that for λn=c ln(| Sn′ |)/| Sn′ | the largest component of random induced subgraphs of Xncontains almost all vertices.