Evolutionary graph colouring

The problem we are addressing in this paper was proposed at SIROCCO'98 by Kranakis, with possible applications to finding a consensus in distributive networks. The problem can be viewed as a simple game on graphs, in a way similar to the game of life. The aim is to colour all vertices of a graph, which are initially coloured with three colours 0, 1, and 2, with a single colour. The only rule is that in one move, two endvertices of an edge can, if they are coloured differently, be re-coloured by the third colour. The question is for which graphs and initial colourings the aim can be achieved and to find an efficient algorithm if it is possible. We solve the problem completely by characterizing the graphs and initial colourings which can be transformed to a monochromatic colouring. Moreover, if the monochromatic colouring exists then it can be found in linear time.