Publication details for Professor Matthew JohnsonBonamy, Marthe, Bousquet, Nicolas, Feghali, Carl & Johnson, Matthew (2019). On a conjecture of Mohar concerning Kempe equivalence of regular graphs. Journal of Combinatorial Theory, Series B 135: 179-199.
- Publication type: Journal Article
- ISSN/ISBN: 0095-8956
- DOI: 10.1016/j.jctb.2018.08.002
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes.
A conjecture of Mohar (2007) asserts that, for , all k-colourings of a k-regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each . We also report the implications of this result on the validity of the Wang–Swendsen–Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.