Cover time of graphs with bounded genus

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 vertices of the graph. It is known that the cover time of any finite connected $n$-vertex graph is at least $(1 + o(1)) n \log n$ and at most $(1 + o(1)) \frac{4}{27} n^3$. By Jonasson and Schramm, the cover time of any bounded-degree finite connected $n$-vertex planar graph 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. In particular, the lower bound is established via the use of circle packing of planar graphs on the Riemann sphere. In this paper, we show that the cover time of any finite $n$-vertex graph $G$ with maximum degree $\Delta$ on the compact Riemann surface $S$ of given genus $g$ is at least $c n(\log n)^2/ \Delta(g + 1)$ and at most $(6 + o(1))n^2$, where $c$ is an absolute constant, if $n$ is sufficiently large and three sufficient conditions for $S$ and a circle packing of $G$ filling $S$.