Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings

This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the (k, q)-colourings of a graph G can be obtained by successively recolouring a single vertex provided k/q ≥ 2col(G) along the lines of Cereceda, van den Heuvel and Johnson’s result for kcolourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for (k, q)-precolourings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.