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) n log n and at most 1 + o(1) 4 27 n 3 . This paper proves that for bounded-degree planar graphs the cover time is at least cn(log n) 2 , and at most 6n 2 , 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.