  We investigate a conjecture on the cover times of planar graphs by means of
large Monte Carlo simulations. The conjecture states that the cover time
$\tau(G_{N})$ of a planar graph $G_{N}$ of $N$ vertices and maximal degree $d$
is lower bounded by $\tau(G_{N}) \geq C_{d} N(\ln N)^2$ with $C_{d} = (d/4\pi)
\tan (\pi/d)$, with equality holding for some geometries. We tested this
conjecture on the regular honeycomb ($d=3$), regular square ($d=4$), regular
elongated triangular ($d=5$), and regular triangular ($d=6$) lattices, as well
as on the nonregular Union Jack lattice ($d_{\rm min}=4$, $d_{\rm max}=8$).
Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as
an equality for most of these lattices, with an interesting issue in the case
of the Union Jack lattice. The data for the honeycomb lattice, however,
violates the bound with the conjectured constant. The empirical probability
distribution function of the cover time for the square lattice is also briefly
presented, since very little is known about cover time probability distribution
functions in general.
