Component structure induced by a random walk on a random graph

We consider random walks on two classes of random graphs and explore the likely structure of the vacant set viz. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs Gn,p (above the connectivity threshold) and for random regular graphs Gr, r ≥ 3 there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. Thus for t ≤ (1 − ǫ)t∗ we have a unique giant plus logarithmic size components and for t ≥ (1 + ǫ)t∗ we only have logarithmic sized components.