Generating even triangulations on the torus

A triangulation on a surface F is a fixed embedding of a loopless graph on F with each face bounded by a cycle of length three. A triangulation is even if each vertex has even degree. We define two reductions for even triangulations on surfaces, called the 4-contraction and the twin-contraction. In this paper, we first determine the complete list of minimal 3-connected even triangulations on the torus with respect to these two reductions. Secondly, allowing a vertex of degree 2 and replacing the twin-contraction with another reduction, called the 2-contraction, we establish the list for all minimal even triangulations on the torus. We also describe several applications of the lists for solving problems on even triangulations.