Tutte Colloquium -Kathie Cameron-Reconfiguration of Vertex Colourings

Abstract: A k-colouring of a graph is an assignment of at most k colours to its vertices so that the ends of each edge get different colours. We consider two types of “reconfiguration steps” for transforming a given k-colouring into a target k-colouring. The first is to change the colour of a vertex to a colour which does not appear on any of the vertices it is adjacent to. We say that a graph G is recolourable if for every k greater than its chromatic number, any k-colouring of G can be transformed into any other by these reconfiguration steps. The second (more general) type of reconfiguration step is Kempe swaps. We call a graph Kempe connected if for every k, any k-colouring can be transformed into any other by Kempe swaps.