Component structure of the vacant set induced by a random walk on a random graph

Authors


Abstract

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 Gn,p (above the connectivity threshold) and for random regular graphs Gr,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 Gn,p and Gr we give the value of t*, and the size of Γ(t). For Gr, 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 Dn,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). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Ancillary