Kempe equivalence of colourings of cubic graphs. (January 2017) - Feghali, Carl; Johnson, Matthew; Paulusma, Daniël
Abstract: Given a graph G = ( V, E ) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k ≥ 3, all k -colourings of connected k -regular graphs that are not complete are Kempe equivalent. We address the case k = 3 by showing that all 3 -colourings of a connected cubic graph G are Kempe equivalent unless G is the complete graph K 4 or the triangular prism.