Extending precolourings of circular cliques

Let G be a graph with circular chromatic number χc(G) = k q . Given P ⊆ V (G) where each component of G[P ] is isomorphic to the circular clique Gk,q, suppose the vertices of P have been precoloured with a (k, q)-colouring. We examine under what conditions one can be assured the precolouring extends to the entire graph. We study sufficient conditions based on k ′ q − k q as well as the distance between precoloured components of G[P ]. In particular, we examine a conjecture of Albertson and West showing the conditions for extendibility are more complex than anticipated in their work.