  Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces
other than 2-sphere. There are eight semi-equivelar maps of types
$\{3^{3},4^{2}\}$, $\{3^{2},4,3,4\}$, $\{6,3,6,3\}$, $\{3^{4},6\}$,
$\{4,8^{2}\}$, $\{3,12^{2}\}$, $\{4,6,12\}$, $\{6,4,3,4\}$ exist on the torus.
In this article we show the existence of Hamiltonian cycle in each
semi-equivelar map on the torus except the map of type $\{3,12^{2}\}$. This
result gives the partial solution to the conjecture which is given by
Gr$\ddot{u}$nbaum \cite{grunbaum} and Nash-Williams \cite{nash williams} that
every 4-connected graph on the torus is Hamiltonian.
