  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 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.
