On enumeration of a class of toroidal graphs

Abstract: We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types ${3^{6}}$, ${4^{4}}$, ${6^{3}}$, ${3^{3}, 4^{2}}$, ${3^{2}, 4, 3, 4}$, ${3, 6, 3, 6}$, ${3^{4}, 6}$, ${4, 8^{2}}$, ${3, 12^{2}}$, ${4, 6, 12}$, ${3, 4, 6, 4}$. We know the classification of the maps of types ${3^{6}}$, ${4^{4}}$, ${6^{3}}$ on the torus. In this article, we attempt to classify maps of types ${3^{3}, 4^{2}}$, ${3^{2}, 4, 3, 4}$, ${3, 6, 3, 6}$, ${3^{4}, 6}$, ${4, 8^{2}}$, ${3, 12^{2}}$, ${4, 6, 12}$, ${3, 4, 6, 4}$ on the torus.