Three Moves on Signed Surface Triangulations

We consider finite triangulations of surfaces with signs attached to the faces. Such a signed triangulation is said to have the Heawood property if, at every vertex x, the sum of the signs of the faces incident to x is divisible by 3. For a triangulation G of the sphere, Heawood signings are essentially equivalent to proper 4-vertex-colorings of G. We introduce three moves on signed surface triangulations which preserve the Heawood property. We then prove that every Heawood signed triangulation of the sphere can be obtained from a Heawood signed triangle by a suitable sequence of our moves.