On the cover time of planar graphs

TL;DR

This paper establishes new bounds on the cover time of bounded-degree planar graphs, showing it grows at least as fast as n(log n)^2 and at most as fast as 6n^2, using circle packings.

Contribution

It provides the first non-trivial lower bound for cover time in bounded-degree planar graphs using geometric methods.

Findings

Lower bound of c n(log n)^2 for cover time

Upper bound of 6n^2 for cover time

Use of circle packings to establish bounds

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 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.