Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings

TL;DR

This paper explores the mixing and extension of circular graph colourings, providing bounds, characterizations, and a new extension theorem for precolourings, advancing understanding of graph homomorphism dynamics.

Contribution

It introduces new bounds for mixing circular colourings, proves an extension theorem for precolourings, and characterizes graphs based on their endomorphism monoids.

Findings

All $(k,q)$-colourings can be obtained via vertex recolouring if $k/q \\geq 2col(G)$.

Provides bounds and discusses their sharpness for circular and classical colourings.

Characterizes graphs where the endomorphism monoid can be generated through mixing.

Abstract

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 $k/q\geq 2col(G)$ 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.