A Reconfigurations Analogue of Brooks' Theorem and its Consequences

TL;DR

This paper extends Brooks' Theorem to graph recoloring, showing that from any k-coloring with k > Δ, a Δ-coloring can be reached through a polynomial number of recolorings unless the graph is a complete graph or an odd cycle, with implications for the structure of the reconfiguration graph.

Contribution

It proves an analogue of Brooks' Theorem for recoloring sequences and analyzes the structure and diameter of the reconfiguration graph for bounded degree graphs.

Findings

Any k > Δ coloring can be transformed into a Δ-coloring with O(n^2) recolorings, barring specific graph classes.

The reconfiguration graph R_{Δ+1}(G) has at most two components, with one possibly having diameter O(n^2).

The results classify the complexity and structure of graph recoloring for graphs with bounded maximum degree.

Abstract

Let $G$ be a simple undirected graph on $n$ vertices with maximum degree~$\Delta$. Brooks' Theorem states that $G$ has a $\Delta$-colouring unless~$G$ is a complete graph, or a cycle with an odd number of vertices. To recolour $G$ is to obtain a new proper colouring by changing the colour of one vertex. We show an analogue of Brooks' Theorem by proving that from any $k$-colouring, $k>\Delta$, a $\Delta$-colouring of $G$ can be obtained by a sequence of $O(n^2)$ recolourings using only the original $k$ colours unless $G$ is a complete graph or a cycle with an odd number of vertices, or $k=\Delta+1$, $G$ is $\Delta$-regular and, for each vertex $v$ in $G$, no two neighbours of $v$ are coloured alike. We use this result to study the reconfiguration graph $R_k(G)$ of the $k$-colourings of $G$. The vertex set of $R_k(G)$ is the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. We prove that for $\Delta\geq 3$, $R_{\Delta+1}(G)$ consists of isolated vertices and at most one further component which has diameter $O(n^2)$. This result enables us to complete both a structural classification and an algorithmic classification for reconfigurations of colourings of graphs of bounded maximum degree.