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On the cover time of planar graphs

Johan Jonasson·Oded Schramm
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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.

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Johan Jonasson, Oded Schramm. 2000-05-18. On the cover time of planar graphs. https://arxiv.org/abs/math/0002034

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