Durham Research Online Deposited in DRO

The k-colouring reconfiguration problem asks whether, for a given graph G, two proper k-colourings α and β of G, and a positive integer `, there exists a sequence of at most ` proper k-colourings of G which starts with α and ends with β and where successive colourings in the sequence differ on exactly one vertex ofG. We give a complete picture of the parameterized complexity of the k-colouring reconfiguration problem for each fixed k when parameterized by `. First we show that the k-colouring reconfiguration problem is polynomial-time solvable for k = 3, settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all k ≥ 4, we show that the k-colouring reconfiguration problem, when parameterized by `, is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.