Building a Larger Class of Graphs for Efficient Reconfiguration of Vertex Colouring

Abstract

A k-colouring of a graph G is an assignment of at most k colours to the vertices of G so that adjacent vertices are assigned different colours. The reconfiguration graph of the k-colourings, R k (G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge in R k (G) if they differ in colour on exactly one vertex. For a k-colourable graph G, we investigate the connectivity and diameter of R k+1 (G). It is known that not all weakly chordal graphs have the property that R k+1 (G) is connected. On the other hand, R k+1 (G) is connected and of diameter O(n 2 ) for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and P 4 -free graphs.

Reconfiguration of vertex colouring and forbidden induced subgraphs

“…They also showed that every 3colourable (P 5 , co-P 5 , C 5 )-free graph is ℓ-mixing and the ℓ-recolouring diameter is at most 2n 2 [7]. Biedl, Lubiw, and Merkel [1] showed that every P 4 -sparse graph is ℓ-mixing and the ℓrecolouring diameter is at most 4n 2 . The last author [11] showed that for all p ≥ 1, there exists a k-colourable weakly chordal graph that is not (k + p)-mixing.…”

“…There are 11 graphs on 4 vertices (see Figure 1). Bonamy and Bousquet [3] showed that every P 4 -free graph G is ℓ-mixing and the ℓ-recolouring diameter is at most 2 • χ(G) • n. Biedl, Lubiw, and Merkel [1] investigated a class of graphs that generalizes P 4 -free graphs. The following theorem was not directly stated in [1] but follows from the proof of Theorem 1 in [1] and improves the bound on the ℓ-recolouring diameter of a P 4 -free graph.…”