An exchange move that preserves tricolorability

Edge-coloring of the connectivity map for the completely foldable triangle grid.

When the edges of the connectivity map are colored the same as the edges of the triangulation (the triangle edges being colored with the mean of the node colors at each end,) left-right, geodesic paths (Petrie paths) cycle through three colors, and facial cycles cycle through two colors.

By shifting branches past each other two-past-two, and then swapping the colors of two edges, a doubled exchange can be made that preserves tricolorability.

An exchange move that preserves tricolorability.