In Most 6-regular Toroidal Graphs All 5-colorings are Kempe Equivalent

TL;DR

This paper proves that in most 6-regular toroidal graphs, all 5-colorings are Kempe equivalent, extending known results for 6-colorings and connecting graph coloring to statistical mechanics.

Contribution

It establishes that all 5-colorings of certain 6-regular toroidal graphs are Kempe equivalent, answering a previously open question for graphs with specific embedding properties.

Findings

All 6-colorings of $T[m\times n]$ are 6-equivalent.

All 5-colorings of $T[m\times n]$ are 5-equivalent for $m,n\ge 6$.

5-colorings are Kempe equivalent in 6-regular graphs with large non-contractible cycles.

Abstract

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.