Recolouring Reed’s conjecture for general graphs

Prove that every graph G is k-recolourable for every integer k satisfying k >= ceil((1/3)omega(G) + (2/3)(Delta(G) + 1)), where k-recolourability means that all proper k-colourings of G are Kempe equivalent.

Background

The paper studies a Kempe-recolouring analogue of Reed's conjecture. It defines k-recolourability by requiring all proper k-colourings of a graph to lie in a single Kempe-equivalence class. A construction of frozen colourings gives an obstruction at the one-third threshold, while the paper proves that below this threshold frozen non-unique colourings cannot occur. The authors therefore conjecture that the one-third coefficient is the exact threshold for all graphs, although other possible obstructions to recolourability would still need to be ruled out.