Geodesic Circulant Graphs Embedded on the Flat Torus

In this paper, we will investigate graphs that arise from the intersections of geodesics embedded on the at torus. For any size collection of geodesics, the number of unique intersections is countable via their slopes. As well, any embedding of two geodesics gives rise to a circulant graph for which its chromatic number can be calculated from their respective slopes. Furthermore, the previously described circulant graphs embedded on the at torus are self-dual. This provides an effective face coloring of any graph arising from the embedding of two slopes on the torus