Recent findings indicate that in most 6-regular toroidal graphs, all 5-colorings are Kempe equivalent, meaning that one can transform any valid 5-coloring into another through a series of Kempe swaps.

Mike Albertson asked if the conjecture is easier for toroidal graphs. He can show that there are 7 matchings together containing every edge exactly twice. He asked if there are 3 matchings that ...