Hamiltonian Cycle in Semi-Equivelar Maps on the Torus

Abstract: 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.