The Coloring Torus of a Graph
Circular colorability geometrizes the study of graph colorings. Using color values in R mod Z, one can reformulate “avoid edge differences close to zero” as “seek edge differences close to 1/2”, then generalize to arbitrary edge difference goals. We define the coloring torus XG of a graph G to be the space of all such coloring problems. XG is made of the same stuff as the color values R mod Z: it is is Rn mod L for a lattice L, so we can apply toric methods to its study. For planar graphs, the observed distances from 0 to torsion points of XG form a pattern generalizing the four color theorem.