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

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. 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. Bonamy, Bousquet, Feghali, and Johnson asked if this holds when $G$ is formed from the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces (this graph is of interest in statistical mechanics). We answer their question affirmatively when $m,n \geq 6$. This is joint work with Reem Mahmoud.

combinatoricscomplex variablesfunctional analysisgeneral mathematicsgroup theoryK-theory and homologynumber theoryoperator algebrasprobabilityquantum algebrarings and algebrasrepresentation theory

Audience: learners

GAG seminar

Series comments: Description: Non-specialized research seminar