  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 the other using Kempe swaps. The triangulated toroidal grid,
$T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product
of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and
Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In
contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of
$T[m\times n]$ are 6-equivalent. They asked whether the same is true for
5-colorings. We answer their question affirmatively when $m,n\ge 6$. Further,
we show that if $G$ is 6-regular with a toroidal embedding where every
non-contractible cycle has length at least 7, then all 5-colorings of $G$ are
5-equivalent. Our results relate to the antiferromagnetic Pott's model in
statistical mechanics.
