Reconfiguration of List Colourings

TL;DR

This work addresses the reconfiguration of proper list colourings on connected graphs by analyzing how local list-size conditions influence global connectivity of the recolouring state space. The authors establish a Key Lemma under private lists where $|L(v)|\ge\deg(v)+1$ for all vertices and at least one vertex has $|L(w)|\ge\deg(w)+2$, proving the recolouring graph is connected with diameter $O(n^2)$ and linking this to ergodicity of Glauber dynamics on unfrozen colourings. They extend these results to a Main Theorem for graphs with maximum degree $\Delta\ge3$, show a sharp Shattering phenomenon when a single list is reduced to $\deg(v)$, and provide a detailed treatment of the regular identical-list case as a local Brooks-type generalisation; they also discuss sharpness, open questions, and the more general correspondence colouring setting. The work offers a local, constructive understanding of phase transitions in recolouring dynamics, with implications for sampling and counting proper colourings under private lists. Overall, it formalises how small, vertex-local changes in colour availability can dramatically alter the global structure of feasible colourings and their recolouring trajectories.

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 $Δ$ 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 $\{1,\ldots,Δ+1\}$.

Figures (4)

• Figure 1: The harder case in the proof of \ref{['3connected-reg-lem']}, and also of Claim 1 in the proof of \ref{['regular-thm']}: Vertex names (left) and key colourings (right) in the reconfiguration sequence from $\alpha$ to $\beta$.
• Figure 2: Left: The general case, guaranteed by Claim 2, when there exists $v\in V(H_2)$ with $S\not\subseteq N(v)$. Right: An exceptional case, when no such $v$ exists, and $G[V(H_1)\cup S]=K_{\Delta+1}-z_1z_2$.
• Figure 3: Proving Claim 3: Vertex names (left) and key colourings in the reconfiguration sequence (right). For simplicity, we show the case when $\alpha(x_1)=1, \beta(y_1)=2, c=3$, and $\{x_1,x_2\}\cap \{y_1,y_2\}=\varnothing$. However, this intersection could be non-empty and/or we might have $\beta(y_1)=\alpha(x_1)$.
• Figure 4: A $4$-fold correspondence-cover $(L,H)$ of $K_4$ for which $\widehat{\mathcal{C}}(K_4,(L,H))$ is not connected.

Theorems & Definitions (64)

• Theorem 1: Feghali, Johnson, and Paulusma FJP
• Conjecture 2
• Remark
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof