Maryam Verdian-Rizi, Toroidal triangulations with few odd degree vertices

We present a simple geometric description for the set of toroidal triangulations with two odd vertices where each vertex has degree five or more. Each such triangulation is described by a cut-and-glue construction starting from an infinite triangular grid. To achieve that, we define some invariants to study the cycles of toroidal triangulations. The motivation for studying such family comes from Fisk triangulations and Grünbaum coloring, which will be discussed as well.