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 {G N,k : N ∈ N * }, together with two k-colourings for each G N,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(2 N ), whereas the size of G N is O(N 2 ). 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.
The problem remains PSPACE-complete for bipartite graphs when k ≥ 4, for planar graphs when 4 ≤ k ≤ 6 and for planar bipartite graphs for k = 4 REF .
Finding Paths between Graph Colourings : PSPACE-completeness and Superpolynomial Distances
Mathematics Computer Science