Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings

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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 k-colourings. 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.