In most 6-regular toroidal graphs all 5-colorings are Kempe equivalent

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two k-colorings are k-equivalent if we can transform one into other using swaps. The triangulated toroidal grid, T[m×n], is formed from (a embedding of) Cartesian product of Cm and Cn by adding parallel diagonals inside all 4-faces. Mohar Salas showed that not 4-colorings T[m×n] 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, Johnson 6-colorings 6-equivalent. They asked whether same true for 5-colorings. We answer their question affirmatively when m,n≥6. Further, show G 6-regular with where every non-contractible cycle has length at least 7, then 5-colorings 5-equivalent. Our results relate to antiferromagnetic Pott’s model statistical mechanics.