Reconfiguration of List Colourings

TL;DR

This paper proves that for connected graphs with maximum degree at least 3, any two proper list colourings can be transformed into each other with a quadratic number of recolouring steps, highlighting a phase transition in colouring dynamics.

Contribution

It generalizes previous results by showing a quadratic bound on recolouring steps for list colourings with lists of size at least degree+1, and identifies a phase transition when list sizes are reduced.

Findings

Recolouring between proper list colourings is possible within O(n^2) steps.

Reducing list size at a vertex can cause the colouring space to shatter.

Results extend and strengthen previous uniform list colouring theorems.

Abstract

Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $\Delta$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,\Delta+1\}$.