Extending precolourings of circular cliques

Abstract

Let G be a graph with circular chromatic number Xe(G) = k/q. Given P ⊆ V (G) where the components of G [P] are isomorphic to the circular clique Gk,g suppose the vertices of P have been precoloured with a (k’,g’) - colouring. We examine under what conditions one can be assured the colouring extends to the entire graph. We stud y 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