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 gra...

• Type: Conference
• Year: 2007
• Where: MFCS
• Authors: Paul S. Bonsma, Luis Cereceda
• Keywords: Bipartite Planar Graphs, MFCS 2007, Planar Graphs, Proper Vertex K-colourings