Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2007-12

May 2007

Finding Paths Between Graph Colourings: PSPACE-completeness and Superpolynomial Distances

Paul Bonsma and Luis Cereceda

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 ∈ Ν}, 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.

A PDF file (214 kB) with the full contents of this report can be downloaded by clicking here.

Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2007-12, together with your name and postal address to:
CDAM Research Reports Series
Centre for Discrete and Applicable Mathematics
London School of Economics
Houghton Street
London WC2A 2AE, U.K.
Phone: +44(0)-20-7955 7494.
Fax: +44(0)-20-7955 6877.
Email: info@maths.lse.ac.uk 

Introduction to the CDAM Research Report Series.
CDAM Homepage.

Copyright London School of Economics & Political Science 2007