#### Date of Original Version

6-20-2011

#### Type

Article

#### Rights Management

This is the accepted version of the article which has been published in final form at http://dx.doi.org/10.1002/rsa.20402

#### Abstract or Description

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(*t*) be the subgraph induced by the vacant set of the walk at step *t*. We show that for random graphs *G*_{n,p} (above the connectivity threshold) and for random regular graphs *G*_{r},*r* ≥ 3, the graph Γ(*t*) undergoes a phase transition in the sense of the well-known ErdJW-RSAT1100590x.png -Renyi phase transition. Thus for *t* ≤ (1 - *ε*)*t*^{*}, there is a unique giant component, plus components of size *O*(log *n*), and for *t* ≥ (1 + *ε*)*t*^{*} all components are of size *O*(log *n*). For *G*_{n,p} and *G*_{r} we give the value of *t*^{*}, and the size of Γ(*t*). For *G*_{r}, we also give the degree sequence of Γ(*t*), the size of the giant component (if any) of Γ(*t*) and the number of tree components of Γ(*t*) of a given size *k* = *O*(log*n*). We also show that for random digraphs *D*_{n,p} above the strong connectivity threshold, there is a similar directed phase transition. Thus for*t* ≤ (1 - *ε*)*t*^{*}, there is a unique strongly connected giant component, plus strongly connected components of size *O*(log *n*), and for *t* ≥ (1 + *ε*)*t*^{*}all strongly connected components are of size *O*(log *n*).

#### DOI

10.1002/rsa.20402

#### Published In

Random Structures and Algorithms, 42, 2, 135-158.