Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k=2, and decidable in polynomial time for k=3. Here we prove it is PSPACE-complete for all k≥4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4≤k≤6, and (ii) bipartite planar graphs and k=4. Moreover, the values of k in (i) and (ii) are tight, in the sense that for larger values of k, it is always possible to recolour α to β.We also exhibit, for every k≥4, a class of graphs {GN,k:N∈N∗}, together with two k-colourings for each GN,k, such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2N), whereas the size of GN is O(N2). This is in stark contrast to the k=3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4≤k≤6 planar graphs and (ii) for k=4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure.