Finding Paths between Colourings

Given a 3-colourable graph G and two proper vertex 3-colourings α and β of G, consider the following question : is it possible to transform α into β by recolouring vertices of G one at a time, making sure that all intermediate colourings are proper 3-colourings? We prove that this question is answerable in polynomial time. We do so by characterising the instances G,α, β where the transformation is possible; the proof of this characterisation is via an algorithm that either finds a sequence of recolourings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolourings does exist, the algorithm uses O(|V (G)|2) recolouring steps and in many cases returns a shortest sequence of recolourings. We also exhibit a class of instances G,α, β that require Ω(|V (G)|2) recolouring steps.