2008) 'Connectedness of the graph of vertex-colourings.', Discrete mathematics., 308 (5-6). pp. 913-919. Further information on publisher's website: http://dx. Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractFor a positive integer k and a graph G, the k-colour graph of G, C k (G), is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k-colourings are joined by an edge in C k (G) if they differ in colour on just one vertex of G. In this note some results on the connectivity of C k (G) are proved. In particular it is shown that if G has chromatic number k ∈ {2, 3}, then C k (G) is not connected. On the other hand, for k ≥ 4 there are graphs with chromatic number k for which C k (G) is not connected, and there are k-chromatic graphs for which C k (G) is connected.

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Given a 3-colorable graph G together with two proper vertex 3-colorings and of G, consider the following question: is it possible to transform into by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, , where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between and , or exhibits a structure which proves that no such

a b s t r a c tSuppose 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.

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