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Hamiltonian Cycle in Semi-Equivelar Maps on the Torus

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.