  A Hamiltonian cycle of a graph is a closed path that visits every vertex once
and only once. It has been difficult to count the number of Hamiltonian cycles
on regular lattices with periodic boundary conditions, e.g. lattices on a
torus, due to the presence of winding modes. In this paper, the exact number of
Hamiltonian cycles on a random trivalent fat graph drawn faithfully on a torus
is obtained. This result is further extended to the case of random graphs drawn
on surfaces of an arbitrary genus. The conformational exponent gamma is found
to depend on the genus linearly.
