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 . 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.