On the cover time of planar graphs

Abstract

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1 + o(1))nlogn and at most (1 + o(1))4/27n3 This paper proves that for bounded-degree planar graphs the cover time is at least cn(log n)2, and at most 6n2, where c is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings. © 2000 Applied probability trust.