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

Abstract

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, even for bipartite graphs, as well as for: (i) planar graphs and 4 ≤ k ≤ 6, and (ii) bipartite planar graphs and k = 4. The values of k in (i) and (ii) are tight. 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. 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. The graphs GN, k can also be taken to be bipartite, as well as (i) planar for 4 ≤ k ≤ 6, and (ii) planar and bipartite for k = 4. This provides a remarkable correspondence between the tractability of the problem and its underlying structure. © Springer-Verlag Berlin Heidelberg 2007.