On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Marthe Bonamy (1) , Nicolas Bousquet (2) , Carl Feghali (3) , Matthew Johnson

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 G-SCOP_OC - Optimisation Combinatoire
3 UiB - University of Bergen

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.

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Marthe Bonamy, Nicolas Bousquet, Carl Feghali, Matthew Johnson. On a conjecture of Mohar concerning Kempe equivalence of regular graphs. Journal of Combinatorial Theory, Series B, 2019, 135, pp.179-199. ⟨10.1016/j.jctb.2018.08.002⟩. ⟨hal-02467364⟩