A Reconfigurations Analogue of Brooks’ Theorem

Abstract

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-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 that from a k-colouring, k > Δ, a Δ-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 = Δ + 1, G is Δ-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. It is known that

• if k ≤ Δ(G), then R _k (G) might not be connected and it is possible that its connected components have superpolynomial diameter,

• if k ≥ Δ(G) + 2, then R _k (G) is connected and has diameter O(n ²).

We complete this structural classification by settling the missing case:

• if k = Δ(G) + 1, then R _k (G) consists of isolated vertices and at most one further component which has diameter O(n ²).

We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R _k (G) by settling the case k = Δ(G) + 1. The problem is

• O(n ²) time solvable for k = 3,

• PSPACE-complete for 4 ≤ k ≤ Δ(G),

• O(n) time solvable for k = Δ(G) + 1,

• O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes).