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 $\bigl(1+o(1)\bigr)n\log n$ and at most $\bigl(1+o(1)\bigr)\frac{4}{27}n^3$. This paper proves that for bounded-degree planar graphs the cover time is at least $c n(\log n)^2$, 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.